# Bio-Inspired Space Robotic Control Compared to Alternatives

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

**Box 1.**Problem Statement

**Problem Statement**: Amongst the many available options for autonomously controlling deep space robots, is one method better than another at simultaneously seeking exact precision, vibration elimination, fuel minimization, and robustness?

#### 1.1. Broad Context and Why This Study Is Important

#### 1.1.1. Cislunar Space

#### 1.1.2. Cislunar Robotic Operations

Variable/Acronym | Definition | Variable/Acronym | Definition |
---|---|---|---|

$\widehat{y}$ | Centerline unit vector | $k$ | Appendage stiffness |

${T}_{C}$ | Control torque | ${m}_{i},{I}_{i}\forall i=2\dots 5$ | Flexible masses and inertias |

${J}_{1}$ | Main body inertia mass moment | ${J}_{2}$ | Flexible inertia mass moment |

${\theta}_{1}$ | Main body rotation angle | ${\theta}_{i},{W}_{i}\forall i=2\dots 5$ | Translation and rotation angles |

^{1}Such tables are offered throughout the manuscript to aid readability.

**Figure 5.**Space robots with cylindrical center rigid bodies and highly flexible appendages. (

**a**) NASA’s first humanoid space robot. Image credit: NASA [3]. (

**b**) Laboratory flexible rotational spacecraft hub with a free-floating, planar air-bearing, very light robotic arm, the schematic of which is displayed in subfigure (

**c**).

#### 1.2. Broad Review of Modeling and Control from First Principles to Modern Instantiations

^{®}/Simulink

^{®}toolbox evolved and was presented in [55], effectively reducing the coding burden of future investigations.

#### 1.3. The Current State of the Research Field and Key References

- Gain stabilization [56]: Tuning of gain to achieve stability of the rigid-body mode. Advantages: simplicity and based on well-known mathematics. Disadvantages: imprecise and uses effort (or equivalently fuel) wastefully compared to more modern methods.
- Classical second-order structural filtering [56]: Second-order filters designed for each chosen resonance and anti-resonance, usually of the lowest mode or the lowest two modes to ensure stability. Advantages: aids fuel usage of gain stabilization approaches. Disadvantages: mathematic model must have precision and remain not time varying.
- Rigid-body, minimum-fuel input trajectory shaping: Apply control analytically derived from constrained control-minimization boundary value problem solutions. Advantages: existing proofs of mathematical optimality. Disadvantages: lacks robustness.
- Single-frequency trajectory shaping: The fashion commanded trajectory from a single sinusoid chosen to avoid mode frequencies of the flexible robot. Advantages: simplicity. Disadvantages: still need accurate mathematical models to properly pick the single frequency.
- Flatten the curve to improve stability: Use option #2 to compensate for all structural modes seeking to create a magnitude response curve resembling such a curve for a second-order rigid-body system (primary motivation remains increased system stability). Advantages: aids fuel usage of gain stabilization approaches. Disadvantages: mathematic model must have precision and remain not time varying.
- Flatten the curve to improve trajectory tracking: This option is like option #7, except choosing parts of modes (resonance or anti-resonance) to minimize trajectory tracking errors (proposed in this manuscript). Advantages: aids fuel usage of gain stabilization approaches. Disadvantages: mathematic model must have precision and remain not time varying.
- Deterministic artificial intelligence: Use physics to define robot self-awareness, while adapting or learning time-varying physical system parameters (e.g., mass, mass moments, stiffness, and damping). Advantages: proofs of optimality, robustness and simple algorithms. Disadvantages: relatively unknown compared to peer methods.
- 9.1
- Self-awareness statements [62]: Use governing equations from physics to exclusively define robot self-awareness, while prescribing necessary trajectories to be tracked (currently only sinusoidal trajectories and control-minimizing trajectories are in the literature).
- 9.2

#### 1.4. Controversial and Diverging Hypotheses—Literature Gaps

#### 1.5. Main Aim of the Work and Highlighting of Principal Conclusions

#### 1.6. Novelties Presented

- Commanded trajectory-shaping options are compared using control effort and tracking accuracy, and recommendations are offered.
- Feedforward controls are compared using control effort and tracking accuracy, and recommendations are offered.
- Commanded trajectories are compared with filtered feedback and no feedforward using least control effort tracking accuracy, and recommendations are offered.
- Mode 1 filtering options are compared using control effort tracking accuracy, and recommendations are offered.
- Mode 3 filtering options are compared using control effort tracking accuracy, and recommendations are offered.
- Mode 4 filtering options are compared using control effort tracking accuracy, and recommendations are offered.
- Overall recommendations are made for selection of commanded trajectories, feedforward controls, and filtered versus unfiltered feedback.
- The least control effort was achieved with step trajectories, rigid-body optimal feedforward control and unfiltered feedback, while recommendations are offered based on tracking accuracy and control effort.

## 2. Materials and Methods

#### 2.1. Space Robot Modeling

**Figure 6.**(

**a**) Laboratory flexible spacecraft robotic arm (very lightweight) attached to a free-floating planar air-bearing rotational hub. Image used in compliance with image use policy [2], “U.S. Department of Defense photographs and imagery, unless otherwise noted, are in the public domain”. (

**b**) Schematic of center-lined, cylindrical, rigid spacecraft. (

**c**) The flexible arm is modeled using the lumped-mass technique, where arm mass is distributed to discretized nodes. Images taken from [60] in compliance with respective image use policies [65].

Variable/Acronym | Definition | Variable/Acronym | Definition |
---|---|---|---|

$\widehat{y}$ | Centerline unit vector | $k$ | Appendage stiffness |

${T}_{C}$ | Control torque | ${m}_{i},{I}_{i}\forall i=2\dots 5$ | Flexible masses and inertias |

${J}_{1}$ | Main body inertia mass moment | ${J}_{2}$ | Flexible inertia mass moment |

${\theta}_{1}$ | Main body rotation angle | ${\theta}_{i},{W}_{i}\forall i=2\dots 5$ | Translation and rotation angles |

^{1}Such tables are offered throughout the manuscript to aid readability.

Variable/Acronym | Definition |
---|---|

${\left(\right)close="|"\; separators="|">F}_{}inertial$, ${\left(\right)close="|"\; separators="|">T}_{}inertial$ | Externally applied force and torque expressed in inertial coordinates |

$F$, $T$ | Externally applied force and torque expressed in non-inertial coordinates |

$m$, $J$ | Body’s mass and mass moment of inertia |

${\left(\right)close="|"\; separators="|">a}_{}inertial={\left(\right)close="|"\; separators="|">\ddot{x}}_{}inertial$ | Resulting accelerations expressed in inertial coordinates |

$\omega $, $\dot{\omega}$ | Angular velocity and acceleration vectors |

${x}_{1},{x}_{2}$; ${\ddot{x}}_{1},{\ddot{x}}_{2}$ | Translational velocity and acceleration vectors |

${k}_{1},{k}_{2}$ | Flexible member stiffnesses |

$\left[M\right],\left[K\right]$ | Assembled matrices of masses and stiffnesses |

^{1}Such tables are offered throughout the manuscript to aid readability.

Variable/Acronym | Definition |
---|---|

${I}_{zz}$ | Body principal moment of inertia with respect to Z-axis |

$\ddot{\theta}$ | Angular acceleration of the system rotation angle, $\theta $ |

$D$ | Rigid–elastic coupling term |

$\ddot{q}$ | Acceleration in generalized displacement coordinates |

${I}_{w}$ | Reaction wheel principal moment of inertia with respect to C, Z axis |

${\ddot{\theta}}_{W}$ | Angular acceleration of the reaction wheel rotation angle, ${\theta}_{W}$ |

$T$ | Control torque of the spacecraft reaction wheel |

${T}_{D}$ | Disturbance torques |

^{1}Such tables are offered throughout the manuscript to aid readability.

#### 2.2. Competing Control Design Methodologies

- Gain stabilization,
- Classical second-order structural filtering,
- Input shaping,
- Whiplash compensation,
- Rigid-body minimum-fuel input trajectory shaping,
- Single-frequency trajectory shaping,
- Flatten the curve to improve stability,
- Flatten the curve to improve trajectory tracking,
- Deterministic artificial intelligence:
- 9.1
- Self-awareness statements, and
- 9.2
- Adaption or optimal learning.

#### 2.3. Selectable Options: Trajectories, Feedforward, Feedback, and Filtering

^{®}including subsystems for the selectable commanded trajectory, selectable feedforward controls, feedback controller, structural filters, and the selection subsystem to activate feedforward, feedback, and structural filtering. Those subsystems are fed to control the flexible space robot’s subsystem in a unit-feedback loop resulting in a displayable rotation angle.

#### 2.3.1. Commanded Trajectories

^{®}subsystem used to select between commanded trajectories is displayed in Figure 8a including ubiquitous step commands, rigid-body control-minimizing optimal commands, whiplash compensation, time-delay input-shaped trajectories, and single sinusoidal commanded trajectories. Figure 8b displays a subsystem used to formulate trajectories that are non-zero only when maneuvering. Meanwhile, Figure 8c,d display notion subsystem outputs.

#### 2.3.2. Feedback Filtering

**Figure 9.**Topology of simulation created in Simulink

^{®}: (

**a**) structural filters; (

**b**) selectable control combination.

**Figure 10.**Feedback filtering for structural resonances and anti-resonances: (

**a**) frequency response plot of unfiltered, PID controlled space robot with decibel frequency on the abscissa and response magnitude on the ordinant; (

**b**) flattened curve, frequency response plot when all four modes are filtered with both bandpass and notch filters with decibel frequency on the abscissa and response magnitude on the ordinant; (

**c**) second-order notch filters optionally applied at resonances with decibel frequency on the abscissa and response magnitude on the ordinant of the right-hand subplot with Real (Re) and Imaginary (Im) parts displayed in the left-hand plot; (

**d**) second-order bandpass filters optionally applied to anti-resonances with decibel frequency on the abscissa and response magnitude on the ordinant of the right-hand subplot with Real (Re) and Imaginary (Im) parts displayed in the left-hand plot.

Variable/ Acronym | Definition |
---|---|

$Im\left(s\right)$ | Imaginary component of transient response |

$Re\left(s\right)$ | Real component of transient response |

${\zeta}_{p}$ | Damping ratio of pole in denominator of Equation (12) |

${\zeta}_{z}$ | Damping ratio of zero in numerator of Equation (12) |

${\omega}_{c}$ | Center frequency of filter placement |

${\omega}_{p}$ | Center frequency of filter pole placement in denominator of Equation (12) |

${\omega}_{z}$ | Center frequency of filter zero placement in numerator of Equation (12) |

dB | Decibels |

$log$ | Base-10 logarithm |

$Output\left(s\right)$ | Displacement or rotation expressed in Laplace domain |

$Input\left(s\right)$ | Control force or torque expressed in Laplace domain |

${K}_{\infty}$ | Steady state gain |

${\varphi}_{max}$ | Maximum phase lead occurring at frequencies determined by ${\zeta}_{z}$ and ${\zeta}_{p}$ |

${K}_{max}$ | Maximum gain occurring when ${\omega}_{p}={\omega}_{p}$ |

^{1}Such tables are offered throughout the manuscript to aid readability.

#### 2.3.3. Feedforward Controls

## 3. Results

#### 3.1. Comparing Commanded Trajectories with Unfiltered Feedback

**Interim summary.**When comparing commanded trajectories, step trajectories surprisingly led to the least control effort, while single-sinusoid trajectories produce the most accurate tracking, with 150% more control effort.

**Bio-inspired whiplash compensation performed essentially as well as time-delayed input shaping.**

#### 3.2. Comparing Feedforward Controls with Unfiltered Feedback

**Interim summary.**When comparing feedforward controls, rigid-body optimal feedforward with step trajectory command surprisingly led to the least control effort, while time-delay input-shaped feedforward with single-sinusoid trajectories commanded produced the most accurate tracking, with 280% more control effort.

#### 3.3. Comparing Commanded Trajectories with Filtered Feedback

**Interim summary.**When comparing commanded trajectories with filtered feedback and no feedforward, step trajectory commands surprisingly led to the least control effort, while rigid-body optimal trajectories achieved an order of magnitude higher accuracy with 2345% more control effort.

#### 3.4. Comparing Mode 1 Filtering with Single-Sinusoidal Trajectories and No Feedforward

^{®}. Quantitative figures of merit correspond to qualitative results in Figure 14′s display of meaningful performance figures of merit.

**Interim summary.**When comparing mode 1 filtering options bandpass only not surprisingly led to the least control effort, but surprisingly also produced the most accurate tracking.

#### 3.5. Comparing Mode 2 Filtering with Single-Sinusoidal Trajectories and No Feedforward

**Interim summary.**When comparing mode 2 filtering options bandpass and notch led to the least control effort, but surprisingly accurate tracking results were inconsistent.

#### 3.6. Comparing Mode 3 Filtering with Single-Sinusoidal Trajectories and No Feedforward

**Interim summary.**When comparing mode 3 filtering options bandpass led to the least control effort and also produced the most accurate tracking accuracy.

#### 3.7. Comparing Mode 4 Filtering with Single-Sinusoidal Trajectories and No Feedforward

**Interim summary.**When comparing mode 4 filtering options bandpass led to the least control effort and also produced the most accurate tracking accuracy.

#### 3.8. Comparing Modes 1–4 Filtering with Single-Sinusoidal Trajectories and No Feedforward

**Interim summary.**When comparing modes 1–4 filtering options bandpass not only led to the least control effort but also produced the most accurate tracking accuracy.

#### 3.9. Comparison of the Best Options Studies

**Interim summary.**The least control effort was achieved with step trajectories, rigid-body optimal feedforward control and unfiltered feedback, and the effort was 75% less than the average. The best mean tracking error was achieved with sinusoidal trajectories, no feedforward, mode 2 notch filtered, and the tracking error mean was 98% better than the average, while the control effort was 330% higher than the minimum available option. The best tracking error deviation was achieved with sinusoidal trajectories, no feedforward, mode 1–4 bandpass filtered, and the tracking error deviation was 80% better than the average, while the control effort was 42% higher than the minimum available option.

## 4. Discussion

- Bio-inspired trajectory shaping (modified from a time-minimization control per [43]) seems confounded in the presence of classical, unfiltered feedback. While the technique performed well, it was not the exemplary option when compared to the multitude of other available options examined.
- When comparing commanded trajectories, step trajectories surprisingly led to the least control effort, while single-sinusoid trajectories produce the most accurate tracking, with 150% more control effort. The bio-inspired whiplash shaping was optimized in the cited literature for minimum time in a feedforward control sense, while this sequel reveals that the solution is not minimum effort (fuel), nor minimum time in the presence of feedback.
- When comparing feedforward controls, rigid-body optimal feedforward with step trajectory command surprisingly led to the least control effort, while time-delay input-shaped feedforward with single-sinusoid trajectories commanded produced the most accurate tracking, with 280% more control effort.
- When comparing commanded trajectories with filtered feedback and no feedforward, step trajectory commands surprisingly led to the least control effort, while rigid-body optimal trajectories achieved an order of magnitude higher accuracy with 2345% more control effort.
- When comparing mode 1 filtering options bandpass alone not surprisingly led to the least control effort, but surprisingly also produced the most accurate tracking.
- When comparing mode 3 filtering options bandpass led to the least control effort and also produced the most accurate tracking accuracy.
- When comparing mode 4 filtering options bandpass led to the least control effort and also produced the most accurate tracking accuracy.
- The least control effort was achieved with step trajectories, rigid-body optimal feedforward control and unfiltered feedback, and the effort was 75% less than the average. The best mean tracking error was achieved with sinusoidal trajectories, no feedforward, mode 2 notch filtered, and the tracking error mean was 98% better than the average, while the control effort was 330% higher than the minimum available option. The best tracking error deviation was achieved with sinusoidal trajectories, no feedforward, mode 1–4 bandpass filtered, and the tracking error deviation was 80% better than the average, while the control effort was 42% higher than the minimum available option.

## 5. Conclusions

#### 5.1. Controversial or Unexpected Results

#### 5.1.1. Best Control Effort

#### 5.1.2. Best Tracking Error Mean

#### 5.1.3. Best Tracking Error Deviation

#### 5.2. Recommended Future Reseach

#### Deterministic Artificial Intelligence

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Elaboration of Modal System Identification on the Flexible Robot System

W2 | θ2 | W3 | θ3 | W4 | θ4 | W5 | θ5 | U6 | θ6 | u7 | θ7 | u8 | θ8 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

W2 | 958.8179 | 0.0000 | −479.409 | 59.9261 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

θ2 | 0.0000 | 19.9754 | −59.9261 | 4.9938 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

W3 | −479.409 | −59.926 | 958.8179 | 0.0000 | −479.409 | 59.9261 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

θ3 | 59.9261 | 4.9938 | 0.0000 | 19.9754 | −59.9261 | 4.9938 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

W4 | 0 | 0 | −479.409 | −59.926 | 958.8179 | 0.0000 | −479.409 | 59.9261 | 0 | 0 | 0 | 0 | 0 | 0 |

θ4 | 0 | 0 | 59.9261 | 4.9938 | 0.0000 | 19.9754 | −59.9261 | 4.9938 | 0 | 0 | 0 | 0 | 0 | 0 |

W5 | 0 | 0 | 0 | 0 | −479.409 | −59.926 | 479.409 | −59.926 | 0 | 0 | 0 | 0 | 0 | 0 |

θ5 | 0 | 0 | 0 | 0 | 59.9261 | 4.9938 | −59.9261 | 19.9754 | −59.9261 | 4.9938 | 0 | 0 | 0 | 0 |

U6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −59.926 | 958.8179 | 0.0000 | −479.409 | 59.9261 | 0 | 0 |

θ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4.9938 | 0.0000 | 19.9754 | −59.9261 | 4.9938 | 0 | 0 |

U7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −479.409 | −59.926 | 958.8179 | 0.0000 | −479.409 | 59.9261 |

θ7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 59.9261 | 4.9938 | 0.0000 | 19.9754 | −59.9261 | 4.9938 |

U8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −479.409 | −59.926 | 479.4089 | −59.926 |

θ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 59.9261 | 4.9938 | −59.9261 | 9.9877 |

^{1}Notice state sequence alternates translation, then rotation at each node.

Mass | W2 | θ2 | W3 | θ3 | W4 | θ4 | W5 | θ5 | U6 | θ6 | u7 | θ7 | u8 | θ8 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

W2 | 0.4761 | 0.0000 | 0.0037 | −0.0002 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

θ2 | 0.0000 | 0.0000 | 0.0002 | −0.0001 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

W3 | 0.0037 | 0.0002 | 4.76 × 10^{−1} | 0.0000 | 0.0037 | −0.0002 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

θ3 | −0.0002 | −0.0001 | 0.0000 | 0.0000 | 0.0002 | −0.0001 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

W4 | 0 | 0 | 0.0037 | 0.0002 | 0.4761 | 0.0000 | 0.0037 | −0.0002 | 0 | 0 | 0 | 0 | 0 | 0 |

θ4 | 0 | 0 | −0.0002 | −0.0001 | 0.0000 | 0.0000 | 0.0002 | −0.0001 | 0 | 0 | 0 | 0 | 0 | 0 |

W5 | 0 | 0 | 0 | 0 | 0.0037 | 0.0002 | 2.63 × 10^{0} | −0.0004 | 0 | 0 | 0 | 0 | 0 | 0 |

θ5 | 0 | 0 | 0 | 0 | −0.0002 | −0.0001 | −0.0004 | 0.0000 | 0.0002 | −0.0001 | 0 | 0 | 0 | 0 |

U6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0002 | 4.76 × 10^{−1} | 0.0000 | 0.0037 | −0.0002 | 0 | 0 |

θ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.0001 | 0.00 × 10^{0} | 0.0000 | 0.0002 | −0.0001 | 0 | 0 |

U7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0037 | 0.0002 | 0.4761 | 0.0000 | 0.0037 | −0.0002 |

θ7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.0002 | −0.0001 | 0.0000 | 0.0000 | 0.0002 | −0.0001 |

U8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0037 | 0.0002 | 4.66 × 10^{−1} | −0.0004 |

θ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | −0.0002 | −0.0001 | −0.0004 | 0.0000 |

^{1}Notice state sequence alternates translation, then rotation at each node. 4.76 × 10

^{−1}and such indicates $4.76\times {10}^{\u20131}$, where the former notation is used to ameliorate table crowding.

1809.5 | 1415.5 | 1042.2 | 774.3 | 596.8 |

478.8 | 410.9 | 54.9 | 43.7 | 30.9 |

15.8 | 10.2 | 0.7 | 2.1 |

^{1}Corresponding to mode shapes in Table 5.

−1 | 3 | 2 | 0 | −3 | −5 | 3 | 1501 | 383 | 1037 | 443 | −692 | 181 | 240 |

−1097 | 3080 | 4481 | 4992 | 4505 | 3173 | −1154 | −4544 | −136 | 2388 | 2221 | 4049 | 1395 | 1669 |

−1 | 1 | −3 | −7 | −5 | 1 | −2 | −1569 | −215 | 204 | 667 | 1425 | −670 | −712 |

−2158 | 4857 | 3958 | 28 | −3814 | −4883 | 2208 | −943 | −2040 | −7064 | −867 | 912 | −2460 | −1864 |

−1 | −1 | −5 | 0 | 7 | 3 | 2 | 1296 | −125 | −1076 | 125 | 995 | −1385 | −1057 |

−3111 | 4495 | −1058 | −4992 | −1185 | 4481 | −3142 | 5806 | 2118 | 368 | −2572 | 4061 | −3204 | −683 |

0 | 0 | 0 | 1 | 1 | −1 | 1 | −99 | 30 | 113 | −105 | 248 | 2247 | 954 |

−3902 | 2199 | −4861 | −47 | 4875 | −2071 | 3937 | −4288 | −3426 | 4504 | 1878 | 4753 | −3652 | 1689 |

0 | −3 | 4 | 0 | −6 | 6 | 1 | 536 | −918 | 135 | 898 | 754 | −946 | 736 |

−4493 | −1062 | −3138 | 4985 | −3062 | −1232 | −4519 | 815 | 2292 | −2423 | 3091 | 891 | −3893 | 3998 |

0 | 2 | 5 | −7 | 7 | −4 | 0 | −294 | 753 | 446 | 880 | 410 | −1936 | 1905 |

−4872 | −3903 | 2129 | 75 | −2232 | 3920 | 4832 | −2471 | 3385 | −210 | −3658 | 3392 | −4013 | 5178 |

−1 | −2 | 2 | −3 | 4 | −5 | −6 | 90 | −261 | 192 | −699 | −732 | −2945 | 3254 |

−5031 | −5052 | 5012 | −5030 | 5062 | −4984 | −4965 | 3580 | −7823 | 3941 | −7649 | −5153 | −4046 | 5502 |

**Figure A1.**Normalized mode shapes (modal coordinates) displayed in physical coordinates with normalized length on the abscissa in meters and displacements in meters on the ordinant, where each mode shape is annotated by a different color.

#### Equations of Motion in the Standard State Space Form

^{®}(code is included in the appendix) generates the mode shapes used to calculate the rigid–elastic coupling terms. The program outputs the flexible system [A], [B], [C], and [D] matrices of the standard state space form. The results are:

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

7 | 0 | 0.099 | −1.064 | −3.382 | −0.736 | −26.635 | 0 | 1.392 × 10^{−4} | −5.066 × 10^{−4} | −3.298 × 10^{−4} | −4.662 × 10^{−5} | −8.609 × 10^{−4} |

8 | 0 | −0.659 | 1.642 | 5.218 | 1.136 | 41.100 | 0 | −9.264 × 10^{−4} | 7.817 × 10^{−4} | 5.088 × 10^{−4} | 7.194 × 10^{−5} | 1.328 × 10^{−3} |

9 | 0 | 0.188 | −6.435 | −6.433 | −1.401 | −50.670 | 0 | 2.649 × 10^{−4} | −3.064 × 10^{−3} | −6.273 × 10^{−4} | −8.869 × 10^{−5} | −1.638 × 10^{−3} |

10 | 0 | 0.025 | −0.270 | −106.024 | −0.187 | −6.755 | 0 | 3.531 × 10^{−5} | −1.285 × 10^{−4} | −1.034 × 10^{−2} | −1.182 × 10^{−5} | −2.183 × 10^{−4} |

11 | 0 | 0.002 | −0.025 | −0.079 | −249.447 | −0.620 | 0 | 3.241 × 10^{−6} | −1.179 × 10^{−5} | −7.677 × 10^{−6} | −1.579 × 10^{−2} | −2.004 × 10^{−5} |

12 | 0 | 0.022 | −0.233 | −0.742 | −0.162 | −962.998 | 0 | 3.055 × 10^{−5} | −1.112 × 10^{−4} | −7.237 × 10^{−5} | −1.023 × 10^{−5} | −3.113 × 10^{−2} |

^{1}Flexible states where base (rigid-body) rotation is controlled. 4.76 × 10

^{−1}and such indicates $4.76\times {10}^{\u20131}$, where the former notation is used to ameliorate table crowding.

## Appendix B. Initialization Function Callbacks for Simulation

clear all; close all; clc; %This block of code establishes the properties of each beam element a=0.0254; b=0.0016; L=0.25; E=72*10^9; I=a*b^3/12; Li=[12 6*L -12 6*L;6*L 4*L^2 -6*L 2*L^2;-12 -6*L 12 -6*L;6*L 2*L^2 -6*L 4*L^2]; k_beam=E*I/L^3*[Li]; rho_beam=2.8*10^3; %Beam density kg/m^3 A_beam=a*b; %Beam cross sectional area mb=rho_beam*A_beam; %Beam mass per unit length %This block creates the empty stiffness matrix [k] k=zeros(14,14); % This block fills in the stiffness matrix components % Row 1 components start at index=1 % Row 2 components start at index=15 k(1,1)=k_beam(3,3)+k_beam(1,1); k(2,1)=k_beam(4,3)+k_beam(2,1); k(1,2)=k_beam(3,4)+k_beam(1,2); k(2,2)=k_beam(4,4)+k_beam(2,2); k(1,3)=k_beam(1,3); k(2,3)=k_beam(2,3); k(1,4)=k_beam(1,4); k(2,4)=k_beam(2,4); % Row 3 components start at index=29 % Row 4 components start at index=43 k(3,1)=k_beam(3,1); k(4,1)=k_beam(4,1); k(3,2)=k_beam(3,2); k(4,2)=k_beam(4,2); k(3,3)=k_beam(3,3)+k_beam(1,1); k(4,3)=k_beam(4,3)+k_beam(2,1); k(3,4)=k_beam(3,4)+k_beam(1,2); k(4,4)=k_beam(4,4)+k_beam(2,2); k(3,5)=k_beam(1,3); k(4,5)=k_beam(2,3); k(3,6)=k_beam(1,4); k(4,6)=k_beam(2,4); % Row 5 components start at index=59 % Row 6 components start at index=73 k(5,3)=k_beam(3,1); k(6,3)=k_beam(4,1); k(5,4)=k_beam(3,2); k(6,4)=k_beam(4,2); k(5,5)=k_beam(3,3)+k_beam(1,1); k(6,5)=k_beam(4,3)+k_beam(2,1); k(5,6)=k_beam(3,4)+k_beam(1,2); k(6,6)=k_beam(4,4)+k_beam(2,2); k(5,7)=k_beam(1,3); k(6,7)=k_beam(2,3); k(5,8)=k_beam(1,4); k(6,8)=k_beam(2,4); % Row 7 components start at index=89 % Row 8 components start at index=103 k(7,5)=k_beam(3,1); k(8,5)=k_beam(4,1); k(7,6)=k_beam(3,2); k(8,6)=k_beam(4,2); k(7,7)=k_beam(3,3); k(8,7)=k_beam(4,3); k(7,8)=k_beam(3,4); k(8,8)=k_beam(4,4)+k_beam(2,2); k(8,9)=k_beam(2,3); k(8,10)=k_beam(2,4); % Row 9 components start at index=120 % Row 10 components start at index=134 k(9,8)=k_beam(3,2); k(10,8)=k_beam(4,2); k(9,9)=k_beam(3,3)+k_beam(1,1); k(10,9)=k_beam(4,3)+k_beam(2,1); k(9,10)=k_beam(3,4)+k_beam(1,2); k(10,10)=k_beam(4,4)+k_beam(2,2); k(9,11)=k_beam(1,3); k(10,11)=k_beam(2,3); k(9,12)=k_beam(1,4); k(10,12)=k_beam(2,4); % Row 11 components start at index=149 % Row 12 components start at index=163 k(11,9)=k_beam(3,1); k(12,9)=k_beam(4,1); k(11,10)=k_beam(3,2); k(12,10)=k_beam(4,2); k(11,11)=k_beam(3,3)+k_beam(1,1); k(12,11)=k_beam(4,3)+k_beam(2,1); k(11,12)=k_beam(3,4)+k_beam(1,2); k(12,12)=k_beam(4,4)+k_beam(2,2); k(11,13)=k_beam(1,3); k(12,13)=k_beam(2,3); k(11,14)=k_beam(1,4); k(12,14)=k_beam(2,4); % Row 13 components start at index=179 % Row 14 components start at index=193 k(13,11)=k_beam(3,1); k(14,11)=k_beam(4,1); k(13,12)=k_beam(3,2); k(14,12)=k_beam(4,2); k(13,13)=k_beam(3,3); k(14,13)=k_beam(4,3); k(13,14)=k_beam(3,4); k(14,14)=k_beam(4,4); %Display stiffness matrix to check k=k; %END STIFFNESS MATRIX. START MASS MATRIX %Assemble individual beam inertia matrix I_beam=ones(1,8); %Creates empty matrix of I’s for eight node points I_beam=[I_beam.*I]; %Fill in matrix values with beam inertia I_beam(1)=0; %First node point inertia = 0 %This block of code creates the individual beam mass matrix “m_beam” mi=[156 22*L 54 -13*L;22*L 4*L^2 13*L -3*L^2;54 13*L 156 -22*L;-13*L -3*L^2 -22*L 4*L^2]; m_beam=mb*L/420*mi; %This block of code establishes the value of each point mass (mp) %and the system point mass matrix (M) mp=0.455; %Point masses, M M=[0 mp mp mp 2*mp mp mp mp]; %Matrix of 8 point masses (0 First point mass) %Creates a 14x14 empty mass matrix [m] m=zeros(14,14); %Fill in the system mass matrix components % Row 1 components start at index=1 % Row 2 components start at index=15 m(1,1)=m_beam(3,3)+m_beam(1,1)+M(2); m(2,1)=m_beam(4,3)+m_beam(2,1); m(1,2)=m_beam(3,4)+m_beam(1,2); m(2,2)=m_beam(4,4)+m_beam(2,2); m(1,3)=m_beam(1,3); m(2,3)=m_beam(2,3); m(1,4)=m_beam(1,4); m(2,4)=m_beam(2,4); % Row 3 components start at index=29 % Row 4 components start at index=43 m(3,1)=m_beam(3,1); m(4,1)=m_beam(4,1); m(3,2)=m_beam(3,2); m(4,2)=m_beam(4,2); m(3,3)=m_beam(3,3)+m_beam(1,1)+M(3); m(4,3)=m_beam(4,3)+m_beam(2,1); m(3,4)=m_beam(3,4)+m_beam(1,2); m(4,4)=m_beam(4,4)+m_beam(2,2); m(3,5)=m_beam(1,3); m(4,5)=m_beam(2,3); m(3,6)=m_beam(1,4); m(4,6)=m_beam(2,4); % Row 5 components start at index=59 % Row 6 components start at index=73 m(5,3)=m_beam(3,1); m(6,3)=m_beam(4,1); m(5,4)=m_beam(3,2); m(6,4)=m_beam(4,2); m(5,5)=m_beam(3,3)+m_beam(1,1)+M(4); m(6,5)=m_beam(4,3)+m_beam(2,1); m(5,6)=m_beam(3,4)+m_beam(1,2); m(6,6)=m_beam(4,4)+m_beam(2,2); m(5,7)=m_beam(1,3); m(6,7)=m_beam(2,3); m(5,8)=m_beam(1,4); m(6,8)=m_beam(2,4); % Row 7 components start at index=89 m(7,5)=m_beam(3,1); m(7,6)=m_beam(3,2); m(7,7)=m_beam(3,3)+3*mb+M(5)+M(6)+M(7)+M(8); m(7,8)=m_beam(3,4); % Row 8 components start at index=103 % Row 9 components start at index=120 m(8,5)=m_beam(4,1); m(9,8)=m_beam(3,2); m(8,6)=m_beam(4,2); m(9,9)=m_beam(3,3)+m_beam(1,1)+M(6); m(8,7)=m_beam(4,3); m(9,10)=m_beam(3,4)+m_beam(1,2); m(8,8)=m_beam(4,4)+m_beam(2,2); m(9,11)=m_beam(1,3); m(8,9)=m_beam(2,3); m(9,12)=m_beam(1,4); m(8,10)=m_beam(2,4); % Row 10 components start at index=134 % Row 11 components start at index=149 m(10,8)=m_beam(4,2); m(11,9)=m_beam(3,1); m(10,9)=m_beam(4,3)+m_beam(2,1); m(11,10)=m_beam(3,2); m(10,10)=m_beam(4,4)+m_beam(2,2); m(11,11)=m_beam(3,3)+m_beam(1,1)+M(7); m(10,11)=m_beam(2,3); m(11,12)=m_beam(3,4)+m_beam(1,2); m(10,12)=m_beam(2,4); m(11,13)=m_beam(1,3); m(11,14)=m_beam(1,4); % Row 12 components start at index=163 m(12,9)=m_beam(4,1); m(12,10)=m_beam(4,2); m(12,11)=m_beam(4,3)+m_beam(2,1); m(12,12)=m_beam(4,4)+m_beam(2,2); m(12,13)=m_beam(2,3); m(12,14)=m_beam(2,4); % Row 13 components start at index=179 % Row 14 components start at index=193 m(13,11)=m_beam(3,1); m(14,11)=m_beam(4,1); m(13,12)=m_beam(3,2); m(14,12)=m_beam(4,2); m(13,13)=m_beam(3,3)+M(8); m(14,13)=m_beam(4,3); m(13,14)=m_beam(3,4); m(14,14)=m_beam(4,4); %Display the system mass matrix to check m=m; %Calculate the natural frequencies and normal modes [NormalModes,EigenValues]=eig(inv(m)*k); NaturalFrequencies=diag(EigenValues^0.5); ModeShapes=NormalModes; %Check Orthogonality like Homework 1 confirm diagonal matrix of 1’s %to satisfy equation 24 on slide 17 OrthoMass=NormalModes’*m*NormalModes; OrthoStiff=NormalModes’*k*NormalModes; StiffCheck=OrthoStiff/EigenValues; Equation24_OrthoCheck=diag(diag(StiffCheck/OrthoMass)); %Spacecraft Radius to be used designating rigid modal coordinate R=0.381; FeeE=NormalModes; %Designate Elastic mode shapes array FeeE Omega=NaturalFrequencies; %Designate variable name ‘Omega’ as natural frequencies %Designate Rigid modal coordinate FeeR FeeR=[R+L 1 R+L*2 1 R+L*3 1 R+L*4 1 -L 1 -L*2 1 -L*3 1]; Di=FeeE’*m*diag(FeeR); %Calculate Rigid-Elastic Coupling Coefficient DiCheck=det(Di); %Confirm Di is singular...det(Di=0) Z=0.0005; Izz=14; w=diag(NaturalFrequencies); %Generate a diagonal matrix of natural frequency Iw=0.0912; Td=0; %Disturbance Torque Tc=0.1; %Control Torque is Iw*qddot_wheel T=Td+Tc; %Total Torque is sum of disturbance and control torques %Start State Space Development NatFreq = diag(EigenValues).^0.5; r = 0.381; %Radius of the wheel (large rigid body) freqs = sqrt(EigenValues); % NatFreq = EigenValues(1:5,1:5); freqs = freqs(1:5,1:5); zeta = 0.0005; %Given damping ratio for all modes Izz = 14; phi_E = NormalModes(1:14,1:5); phi_R = [r+L,1,r+2*L,1,r+3*L,1,r+4*L,1,-L,1,-2*L,1,-3*L,1]’; M_II = m; Di = [phi_E’*M_II*phi_R]; M_state = [Izz Di’; Di eye(5)]; C_damp = [zeros(6,6)]; C_damp(2:6,2:6) = 2*zeta*freqs; K=[zeros(6,6)]; K(2:6,2:6) = NatFreq; A = [zeros(6),eye(6,6); -inv(M_state)*K, -inv(M_state)*C_damp]; Bprime = [1;0;0;0;0;0]; B = [0 0 0 0 0 0 (inv(M_state)*Bprime)’]’; C = zeros(12,12); C(1,1)=1; D = zeros(12,1); [Gnum,Gden] = ss2tf(A,B,C,D); G1 = tf(Gnum(1,:),Gden) %Manually input Transfer Function to check NUM=[1.998e-015 0.1268 0.007582 166.9 5.591 4.718e004 771 3.412e006 1.218e004 1.576e007 1.475e004 7.11e006]; DEN=[1 0.06125 1326 46.15 3.781e005 6683 2.808e007 1.388e005 1.813e008 2.065e005 9.954e007 0 0]; G=tf(NUM,DEN); %Put PID controller Transfer function into workspace It=14; Z=0.516931; Bandwidth=4; wn=Bandwidth; T=10/Z/wn; Kd=2*Z*wn*It+It/T; Kp=wn^2+2*Z*wn/T; Ki=wn^2/T; PID=tf([Kd Kp Ki],[0 1 0]); %DESIGN FILTERS TO SMOOTH OUT MODE 1 %Design Bandpass filter for w = 10^-0.1478 = 0.711541 Hz wz=0.711541;Zz=0.1;wp=wz;Zp=0.0005; BP1=tf([1/wz^2 2*Zz/wz 1],[1/wp^2 2*Zp/wp 1]); PID_BP1=PID*BP1; %Design Notch filter for w = 10^-0.109 = 0.778037 Hz wz=0.778037;Zz=0.0005;wp=wz;Zp=0.1; Notch1=tf([1/wz^2 2*Zz/wz 1],[1/wp^2 2*Zp/wp 1]); Mode_1=PID*BP1*Notch1; %DESIGN FILTERS TO SMOOTH OUT MODE 2 %Design Bandpass filter for w = 10^0.3223 wz=10^0.3223;Zz=0.1;wp=wz;Zp=0.0005; BP2=tf([1/wz^2 2*Zz/wz 1],[1/wp^2 2*Zp/wp 1]); %Design Notch filter for w = 10^0.405 wz=10^0.405;Zz=0.0006;wp=wz;Zp=0.1; Notch2=tf([1/wz^2 2*Zz/wz 1],[1/wp^2 2*Zp/wp 1]); Mode_2=Mode_1*BP2*Notch2; %Design Lead filter for wz~1, wp~3 %wz=1;Zz=1;wp=3;Zp=1; %Lead=tf([1/wz^2 2*Zz/wz 1],[1/wp^2 2*Zp/wp 1]); %Mode_2=Mode_2*Lead; %DESIGN FILTERS TO SMOOTH OUT MODE 3 %Design Bandpass filter for w = 10^1.0110 wz=10^1.0110;Zz=0.1;wp=wz;Zp=0.0005; BP3=tf([1/wz^2 2*Zz/wz 1],[1/wp^2 2*Zp/wp 1]); %Design Notch filter for w = 10^1.0128 wz=10^1.0128;Zz=0.0005;wp=wz;Zp=0.1; Notch3=tf([1/wz^2 2*Zz/wz 1],[1/wp^2 2*Zp/wp 1]); Mode_3=Mode_2*BP3*Notch3; %DESIGN FILTERS TO SMOOTH OUT MODE 4 %Design Bandpass filter for w = 10^1.49035 wz=10^1.49035;Zz=0.1;wp=wz;Zp=0.0005; BP4=tf([1/wz^2 2*Zz/wz 1],[1/wp^2 2*Zp/wp 1]); %Design Notch filter for w = 10^1.492 wz=10^1.492;Zz=0.0005;wp=wz;Zp=0.1; Notch4=tf([1/wz^2 2*Zz/wz 1],[1/wp^2 2*Zp/wp 1]); Mode_4=Mode_3*BP4*Notch4; %CALCULATE SYSTEM NATURAL FREQUENCIES [NaturalFrequencies,Damping,EigenValue]=damp(G); NaturalFrequencies=NaturalFrequencies; |

## Appendix C. Stop Function Callbacks for Simulation

[mag1,phase1,wout1] = bode(G); Mag1=20*log10(mag1(:)); Phase1=phase1(:); [mag2,phase2,wout2] = bode(G*PID); Mag2=20*log10(mag2(:)); Phase2=phase2(:); [mag3,phase3,wout3] = bode(G*PID*BP1); Mag3=20*log10(mag3(:)); Phase3=phase3(:); [mag4,phase4,wout4] = bode(G*PID*BP1*Notch1); Mag4=20*log10(mag4(:)); Phase4=phase4(:); [mag5,phase5,wout5] = bode(G*PID*BP1*Notch1*BP2); Mag5=20*log10(mag5(:)); Phase5=phase5(:); [mag6,phase6,wout6] = bode(G*PID*BP1*Notch1*BP2*Notch2); Mag6=20*log10(mag6(:)); Phase6=phase6(:); [mag7,phase7,wout7] = bode(G*PID*BP1*Notch1*BP2*Notch2*BP3); Mag7=20*log10(mag7(:)); Phase7=phase7(:); [mag8,phase8,wout8] = bode(G*PID*BP1*Notch1*BP2*Notch2*BP3*Notch3); Mag8=20*log10(mag8(:)); Phase8=phase8(:); [mag9,phase9,wout9] = bode(G*PID*BP1*Notch1*BP2*Notch2*BP3*Notch3*BP4); Mag9=20*log10(mag9(:)); Phase9=phase9(:); [mag10,phase10,wout10] = bode(G*PID*BP1*Notch1*BP2*Notch2*BP3*Notch3*BP4*Notch4); Mag10=20*log10(mag10(:)); Phase10=phase10(:); figure(1); hold on; semilogx(wout1,Mag1,‘--’,‘LineWidth’,1); semilogx(wout2,Mag2,‘LineWidth’,1); semilogx(wout3,Mag3,‘--’,‘LineWidth’,3); semilogx(wout4,Mag4,‘:’,‘LineWidth’,2); hold off; grid on; axis([0.5,40, -100, 150 ]); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); legend(‘Flexible space robot’,‘PID’,‘PID + Bandpass’,‘PID + Notch + Bandpass’) figure(2); hold on; set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); semilogx(wout1,Phase1,’--’,‘LineWidth’,1); semilogx(wout2,Phase2,’LineWidth’,1); semilogx(wout3,Phase3,’--’,‘LineWidth’,3); semilogx(wout4,Phase4,’:’,‘LineWidth’,2); hold off; grid on; figure(3); hold on; semilogx(wout2,Mag2,’--’,‘LineWidth’,1); semilogx(wout4,Mag4,’LineWidth’,1); semilogx(wout5,Mag5,’--’,‘LineWidth’,3); semilogx(wout6,Mag6,’:’,‘LineWidth’,2); hold off; grid on; axis([0.5,40, -100, 150 ]); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); legend(‘PID controlled Flexible space robot’,‘PID+Mode 1’,‘PID + Mode 1 + Bandpass’,‘PID + Mode 1 + Notch + Bandpass’) figure(4); hold on; set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); semilogx(wout2,Phase2,’--’,‘LineWidth’,1); semilogx(wout4,Phase4,’LineWidth’,1); semilogx(wout5,Phase5,’--’,‘LineWidth’,3); semilogx(wout6,Phase6,’:’,‘LineWidth’,2); hold off; grid on; figure(5); hold on; semilogx(wout2,Mag2,’--’,‘LineWidth’,1); semilogx(wout4,Mag4,’LineWidth’,1); semilogx(wout6,Mag6,’--’,‘LineWidth’,3); semilogx(wout7,Mag7,’:’,‘LineWidth’,2); semilogx(wout8,Mag8,’:’,‘LineWidth’,2); hold off; grid on; axis([0.5,40, -100, 150 ]); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); legend(‘PID controlled Flexible space robot’,‘PID + Mode 1’,‘PID + Mode 2’,‘PID + Mode 1 + Mode 2 + Bandpass’,‘PID + Mode 1 + Mode 2 + Bandpass + Notch’) figure(6); hold on; set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); semilogx(wout2,Phase2,’--’,‘LineWidth’,1); semilogx(wout4,Phase4,’LineWidth’,1); semilogx(wout6,Phase6,’--’,‘LineWidth’,3); semilogx(wout7,Phase7,’:’,‘LineWidth’,2); semilogx(wout8,Phase8,’:’,‘LineWidth’,2); hold off; grid on; figure(7); hold on; semilogx(wout2,Mag2,’--’,‘LineWidth’,1); semilogx(wout4,Mag4,’LineWidth’,1); semilogx(wout6,Mag6,’--’,‘LineWidth’,3); semilogx(wout8,Mag8,’:’,‘LineWidth’,2); semilogx(wout9,Mag9,’:’,‘LineWidth’,2); semilogx(wout10,Mag10,’:’,‘LineWidth’,2); hold off; grid on; axis([0.5,40, -100, 150 ]); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); legend(‘PID controlled Flexible space robot’,‘PID + Mode 1’,‘PID + Mode 2’,‘PID + Mode 1 + Mode 2 + Mode 3’,‘PID + Mode 1 + Mode 2 + Mode 3 + Bandpass + Notch’) figure(8); hold on; set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); semilogx(wout2,Phase2,’--’,‘LineWidth’,1); semilogx(wout4,Phase4,’LineWidth’,1); semilogx(wout6,Phase6,’--’,‘LineWidth’,3); semilogx(wout8,Phase8,’:’,‘LineWidth’,2); semilogx(wout9,Phase9,’:’,‘LineWidth’,2); semilogx(wout10,Phase10,’:’,‘LineWidth’,2); hold off; grid on; sys1=G*PID/(1+G*PID); sys2=(G*PID*BP1/(1+G*PID*BP1)); sys3=(G*PID*BP1*Notch1/(1+G*PID*BP1*Notch1)); sys4=(G*PID*BP1*Notch1*BP2/(1+G*PID*BP1*Notch1*BP2)); sys5=(G*PID*BP1*Notch1*BP2*Notch2/(1+G*PID*BP1*Notch1*BP2*Notch2)); sys6=(G*PID*BP1*Notch1*BP2*Notch2*BP3/(1+G*PID*BP1*Notch1*BP2*Notch2*BP3)); sys7=(G*PID*BP1*Notch1*BP2*Notch2*BP3*Notch3/(1+G*PID*BP1*Notch1*BP2*Notch2*BP3*Notch3)); sys8=(G*PID*BP1*Notch1*BP2*Notch2*BP3*Notch3*BP4/(1+G*PID*BP1*Notch1*BP2*Notch2*BP3*Notch3*BP4)); sys9=(G*PID*BP1*Notch1*BP2*Notch2*BP3*Notch3*BP4*Notch4/(1+G*PID*BP1*Notch1*BP2*Notch2*BP3*Notch3*BP4*Notch4)); figure(9); step(sys1,sys2); legend(‘PID’,‘PID + BP1’);set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); figure(10); step(sys1,sys3); legend(‘PID’,‘PID + BP1+Notch1’); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); figure(11); step(sys1,sys4); legend(‘PID’,‘PID + Mode 1 + BP2’); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); figure(12); step(sys1,sys5); legend(‘PID’,‘PID + Mode 1 + BP2 + Notch 2’); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); figure(13); step(sys1,sys6); legend(‘PID’,‘PID + Mode 1 + Mode 2 + BP3’); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); figure(14); step(sys1,sys7); legend(‘PID’,‘PID + Mode 1 + Mode 2 + BP3 + Notch3’); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); figure(15); step(sys1,sys8); legend(‘PID’,‘PID + Mode 1 + Mode 2 + Mode 3 + BP4’); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); figure(16); step(sys1,sys9); legend(‘PID’,‘PID + Mode 1 + Mode 2 + Mode 3 + BP4 + Notch4’); set(gca, ‘FontSize’,28, ‘FontName’,‘Palatino Linotype’); |

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**Figure 1.**Space robots may be represented as cylindrical center rigid bodies and highly flexible appendages. (

**a**) A U.S. Naval Academy pitcher throws to home plate at a baseball tournament (image credit: Technical Sergeant David W. Carbajal) [1,2]. (

**b**) NASA’s first humanoid space robot (image credit: NASA) [3,4].

**Figure 4.**NASA Robotic Refueling Mission (RRM) task: refueling. Individual pieces of hardware show the seals that typical satellite fuel valves have. (

**a**) A tertiary cap with a “lock wire” visible underneath; (

**b**) a safety cap/actuation nut with a securing lock wire; (

**c**) an exposed fuel valve; (

**d**) a safety cap tool removing a safety cap; and (

**e**) a nozzle tool being connected to the now exposed fuel valve, enabling fuel transfer (image credit: NASA [4,18]).

**Figure 8.**Topology of simulation created in Simulink

^{®}: (

**a**) selectable command trajectory; (

**b**) selectable feedforward; (

**c**) selectable commanded trajectories: the thick solid black line is the unit step function, the rigid-body minimum fuel is the thin solid green line,

**bio–inspired whiplash trajectory**is the red dashed line, and the single sinusoidal commanded trajectory is the dotted blue line. (

**d**) Feedforward controls: minimum fuel optimal feedforward is the dashed red line, while time-delay input-shaped feedforward is the solid blue line in the inset plot.

**Figure 11.**Trajectory command comparative simulation experiments performed in Simulink

^{®}with normalized time on the abscissae. The solid blue line indicates step trajectory, no feed-forward, unfiltered; the solid black line indicates whiplash trajectory, no feedforward, unfiltered; the red dotted line indicates time-delayed input-shaped trajectory, no feedforward, unfiltered, where the ordinants display: (

**a**) control in [Newton meters], (

**b**) tracking error in [degrees], and (

**c**) rotation angle in [degrees]. Qualitative results correspond to quantitative figures of merit in Table 6. Rigid-body minimum-fuel input trajectory shaping performed so poorly as to not be presentable.

**Figure 12.**Selectable feedforward comparative simulation experiments performed in Simulink

^{®}with normalized time on the abscissae. The solid blue line indicates step trajectory, rigid-body optimal feed-forward, unfiltered; the solid black line indicates step trajectory, time-delay input shaping feedforward, unfiltered; the red dotted line indicates single-sine trajectory, time-delay input shaping feedforward, unfiltered, where the ordinants display: (

**a**) control in [Newton meters], (

**b**) tracking error in [degrees], and (

**c**) rotation angle in [degrees]. Qualitative results correspond to quantitative figures of merit in Table 7. Single-sinusoid trajectory, rigid-body optimal feedforward, unfiltered performed so poorly as to not be presentable.

**Figure 13.**Selectable commanded trajectory comparative simulation experiments performed in Simulink

^{®}with normalized time on the abscissae. The solid blue line indicates step trajectory, no feedforward, filtered feedback; the solid black line indicates rigid-body optimal trajectory, no feedforward, filtered feedback; the red dotted line indicates whiplash trajectory, no feedforward, filtered feedback; and the green dashed line indicates single-sine trajectory, no feedforward, filtered feedback, where the ordinants display: (

**a**) control in [Newton meters], (

**b**) tracking error in [degrees], and (

**c**) rotation angle in [degrees]. Qualitative results correspond to quantitative figures of merit in Table 8.

**Figure 14.**Selectable mode 1 feedback filtering comparative simulation experiments performed in Simulink

^{®}with normalized time on the abscissae. The solid black line indicates single-sine trajectory, no feedforward, mode 1 bandpass filtered; the solid blue line indicates single-sine trajectory, no feedforward, mode 1 notch filtered; the red dotted line indicates single-sine trajectory, no feedforward, mode 1 bandpass and notch filtered, where the ordinants display: (

**a**) control in [Newton meters], (

**b**) tracking error in [degrees], and (

**c**) rotation angle in [degrees]. Qualitative results correspond to quantitative figures of merit in Table 8.

**Figure 15.**Selectable mode 2 feedback filtering comparative simulation experiments performed in Simulink

^{®}with normalized time on the abscissae. The solid black line indicates single-sine trajectory, no feedforward, mode 2 bandpass filtered; the solid blue line indicates single-sine trajectory, no feedforward, mode 2 notch filtered; the red dotted line indicates single-sine trajectory, no feedforward, mode 2 bandpass and notch filtered, where the ordinants display: (

**a**) control in [Newton meters], (

**b**) tracking error in [degrees], and (

**c**) rotation angle in [degrees]. Qualitative results correspond to quantitative figures of merit in Table 10.

**Figure 16.**Selectable mode 3 feedback filtering comparative simulation experiments performed in Simulink

^{®}with normalized time on the abscissae. The solid black line indicates single-sine trajectory, no feedforward, mode 3 bandpass filtered; the solid blue line indicates single-sine trajectory, no feedforward, mode 3 notch filtered; the red dotted line indicates single-sine trajectory, no feedforward, mode 3 bandpass and notch filtered, where the ordinants display: (

**a**) control in [Newton meters], (

**b**) tracking error in [degrees], and (

**c**) rotation angle in [degrees]. Qualitative results correspond to quantitative figures of merit in Table 10.

**Figure 17.**Selectable mode 4 feedback filtering comparative simulation experiments performed in Simulink

^{®}with normalized time on the abscissae. The solid black line indicates single-sine trajectory, no feedforward, mode 4 bandpass filtered; the solid blue line indicates single-sine trajectory, no feedforward, mode 4 notch filtered; the red dotted line indicates single-sine trajectory, no feedforward, mode 4 bandpass and notch filtered, where the ordinants display: (

**a**) control in [Newton meters], (

**b**) tracking error in [degrees], and (

**c**) rotation angle in [degrees]. Qualitative results correspond to quantitative figures of merit in Table 12.

**Figure 18.**Selectable modes 1–4 feedback filtering comparative simulation experiments performed in Simulink

^{®}with normalized time on the abscissae. The solid black line indicates single-sine trajectory, no feedforward, modes 1–4 bandpass filtered; the solid blue line indicates single-sine trajectory, no feedforward, modes 1–4 notch filtered, where the ordinants display: (

**a**) control in [Newton meters], (

**b**) tracking error in [degrees], and (

**c**) rotation angle in [degrees]. Qualitative results correspond to quantitative figures of merit in Table 13.

**Figure 19.**Space robots with cylindrical center rigid bodies and highly flexible appendages. (

**a**) NASA’s first humanoid space robot. Image credit: NASA [3]. (

**b**) Laboratory flexible rotational spacecraft hub with a free-floating, planar air-bearing, very light robotic arm, the schematic of which is displayed in subfigure (

**c**).

**Table 6.**Trajectory command comparative simulation experiments performed in Simulink

^{®}. Quantitative figures of merit correspond to qualitative results in Figure 11.

Control Methods ^{1} | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Step trajectory, no feedforward, unfiltered | 0.27662 | 0.025967 | 0.29883 |

Bio-inspired whiplash trajectory, no feedforward, unfiltered | 1.0997 | –0.026376 | 0.2936 |

Time-delayed input-shaped trajectory, no feedforward, unfiltered | 1.0997 | –0.026376 | –0.27936 |

Single-sinusoid trajectory, no feedforward, unfiltered | 0.69228 | 0.00052658 | 0.025702 |

^{1}Rigid-body minimum-fuel input trajectory shaping performed so poorly as to not be presentable.

**Table 7.**Selectable feedforward comparative simulation experiments performed in Simulink

^{®}. Quantitative figures of merit correspond to qualitative results in [degrees].

Control Methods ^{1} | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Step trajectory, rigid-body optimal feedforward, unfiltered | 0.15916 | 0.030193 | 0.29752 |

Step trajectory, time-delay input-shaped feedforward, unfiltered | 0.211539 | 0.026717 | 0.27847 |

Single-sinusoid trajectory, rigid-body optimal feedforward, unfiltered | 294.3845 | 6.1359 | 3.9368 |

Single-sinusoid trajectory, time-delay input-shaped feedforward, unfiltered | 0.61639 | –0.0014197 | 0.026812 |

^{1}Rigid-body minimum-fuel input trajectory shaping performed so poorly as to not be presentable.

**Table 8.**Selectable commanded trajectories with filtered feedback (and no feedforward) comparative simulation experiments performed in Simulink

^{®}. Quantitative figures of merit correspond to qualitative results in Table 8.

Control Methods ^{1} | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Step trajectory, no feedforward, filtered feedback | 0.028957 | 0.026908 | 0.30033 |

Rigid-body optimal trajectory, no feedforward, filtered feedback | 3.0251 | 0.0020571 | 0.040348 |

Bio-inspired whiplash, no feedforward, filtered feedback | 0.70804 | –0.023355 | 0.27816 |

Single-sine trajectory, no feedforward, filtered feedback | 0.70804 | –0.023355 | 0.27816 |

^{1}Rigid-body minimum-fuel input trajectory shaping performed so poorly as to not be presentable.

**Table 9.**Selectable mode 1 feedback filtering comparative simulation experiments performed in Simulink

^{®}. Quantitative figures of merit correspond to qualitative results in Figure 14.

Single-Sine Trajectory, No Feedforward, Iterated Feedback Filtering | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Mode 1 bandpass filtered | 0.62207 | 0.00046139 | 0.018780 |

Mode 1 notch filtered | 0.89711 | 0.00046872 | 0.029906 |

Mode 1 Bandpass and notch filtered | 0.71577 | 0.00058161 | 0.020662 |

**Table 10.**Selectable mode 2 feedback filtering comparative simulation experiments performed in Simulink

^{®}. Quantitative figures of merit correspond to qualitative results in Figure 15.

Single-Sine Trajectory, No Feedforward, Iterated Feedback Filtering | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Mode 2 bandpass filtered | 0.47683 | 0.00026452 | 0.022876 |

Mode 2 notch filtered | 0.68856 | 0.00013103 | 0.023599 |

Mode 2 Bandpass and notch filtered | 0.41301 | 0.00016559 | 0.023188 |

**Table 11.**Selectable mode 3 feedback filtering comparative simulation experiments performed in Simulink

^{®}. Quantitative figures of merit correspond to qualitative results in Figure 16.

Single-Sine Trajectory, No Feedforward, Iterated Feedback Filtering | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Mode 3 bandpass filtered | 0.67211 | 0.00015387 | 0.023116 |

Mode 3 notch filtered | 0.68804 | 0.00025072 | 0.023377 |

Mode 3 Bandpass and notch filtered | 0.68986 | 0.00017818 | 0.023274 |

**Table 12.**Selectable mode 4 feedback filtering comparative simulation experiments performed in Simulink

^{®}. Quantitative figures of merit correspond to qualitative results in Figure 12.

Single-Sine Trajectory, No Feedforward, Iterated Feedback Filtering | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Mode 4 bandpass filtered | 0.65649 | 0.00075748 | 0.023116 |

Mode 4 notch filtered | 0.69137 | 0.00330010 | 0.023697 |

Mode 4 Bandpass and notch filtered | 0.68994 | 0.00094136 | 0.023266 |

**Table 13.**Selectable mode feedback filtering comparative simulation experiments performed in Simulink

^{®}. Quantitative figures of merit correspond to qualitative results in Figure 13.

Single-Sine Trajectory, No Feedforward, Iterated Feedback Filtering | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Modes 1–4 bandpass filtered | 0.22672 | 0.0010466 | 0.017807 |

Modes 1–4 notch filtered | 0.90941 | 0.0038787 | 0.030924 |

Control Methods ^{1} | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Bio-inspired whiplash trajectory, no feedforward, unfiltered feedback | 1.0997 | –0.026376 | 0.2936 |

Bio-inspired whiplash, no feedforward, filtered feedback | 0.70804 | –0.023355 | 0.27816 |

Rigid-body optimal trajectory, no feedforward, filtered feedback | 3.0251 | 0.0020571 | 0.040348 |

Time-delayed input-shaped trajectory, no feedforward, unfiltered | 1.0997 | –0.026376 | –0.27936 |

Step trajectory, no feedforward, unfiltered | 0.27662 | 0.025967 | 0.29883 |

Step trajectory, no feedforward, filtered feedback | 0.028957 | 0.026908 | 0.30033 |

Step trajectory, rigid-body optimal feedforward, unfiltered | 0.15916 | 0.030193 | 0.29752 |

Single-sinusoid trajectory, no feedforward, unfiltered | 0.69228 | 0.00052658 | 0.025702 |

Single-sinusoid trajectory, time-delay input-shaped feedforward, unfiltered | 0.61639 | –0.0014197 | 0.026812 |

Sinusoidal trajectories, no feedforward, mode 1 bandpass filtered | 0.62207 | 0.00046139 | 0.018780 |

Sinusoidal trajectories, no feedforward, mode 2 bandpass filtered | 0.47683 | 0.00026452 | 0.022876 |

Sinusoidal trajectories, no feedforward, mode 2 notch filtered | 0.68856 | 0.00013103 | 0.023599 |

Sinusoidal trajectories, no feedforward, mode 2 bandpass and notch filtered | 0.41301 | 0.00016559 | 0.023188 |

Sinusoidal trajectories, no feedforward, mode 3 bandpass filtered | 0.67211 | 0.00015387 | 0.023116 |

Sinusoidal trajectories, no feedforward, mode 4 bandpass filtered | 0.65649 | 0.00075748 | 0.023116 |

Sinusoidal trajectories, no feedforward, mode 1–4 bandpass filtered | 0.22672 | 0.0010466 | 0.017807 |

Average | 0.658023 | 0.007386 | 0.087848 |

^{1}Rigid-body minimum-fuel input trajectory shaping performed so poorly as to not be presentable.

Control Methods ^{1} | Control Effort | Tracking Error Mean | Tracking Error Deviation |
---|---|---|---|

Single-sinusoid trajectory, time-delay input-shaped feedforward, unfiltered | −6% | −119% | −69% |

Step trajectory, rigid-body optimal feedforward, unfiltered | −96% | 264% | 242% |

Sinusoidal trajectories, no feedforward, mode 1–4 bandpass filtered | −66% | −86% | −80% |

^{1}Rigid-body minimum-fuel input trajectory shaping performed so poorly as to not be presentable.

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Sands, T.
Bio-Inspired Space Robotic Control Compared to Alternatives. *Biomimetics* **2024**, *9*, 108.
https://doi.org/10.3390/biomimetics9020108

**AMA Style**

Sands T.
Bio-Inspired Space Robotic Control Compared to Alternatives. *Biomimetics*. 2024; 9(2):108.
https://doi.org/10.3390/biomimetics9020108

**Chicago/Turabian Style**

Sands, Timothy.
2024. "Bio-Inspired Space Robotic Control Compared to Alternatives" *Biomimetics* 9, no. 2: 108.
https://doi.org/10.3390/biomimetics9020108