Minimum-Energy Trajectory Planning for an Underactuated Serial Planar Manipulator

Abstract

Underactuated robotic systems are appealing for industrial use due to their reduced actuator number, which lowers energy consumption and system complexity. Underactuated systems are, however, often affected by residual vibrations. This paper addresses the challenge of generating energy-optimal trajectories while imposing theoretical null residual (and yet practical low) vibration in underactuated systems. The trajectory planning problem is cast as a constrained optimal control problem (OCP) for a two-degree-of-freedom revolute–revolute planar manipulator. The proposed method produces energy-efficient motion while limiting residual vibrations under motor torque limitations. Experiments compare the proposed trajectories to input shaping techniques (ZV, ZVD, NZV, NZVD). Results show energy savings that range from 12% to 69% with comparable and negligible residual oscillations.

1. Introduction

Ensuring an efficient use of resources is the key element that defines the profitability of any manufacturing process. One key element is cost reduction, which is inevitably linked to the cost of energy, which can vary sensibly over time and without a predictable pattern. Beyond direct economic costs, any manufacturing process is also associated with an environmental cost, which is proportional in part to its energy expenditure. This fact has also been recognized at the political level, as demonstrated by the adoption of the Sustainable Development Goals (SDGs) by the EU [1]. In particular, energy reduction is central in both SDG12 and SDG9. The effort to push cleaner production through SDGs is, however, just a part of a bigger and long-lasting effort towards energy efficiency that started 40 years ago [2].

The literature on methods to improve the energy efficiency of automation and robotic systems, i.e., two staples of modern energy production, is rather extended, as testified by the review work [3] that analyzes several methods to achieve the goal of a greener mechatronics [4]. An energy-efficient automatic system is one that implements some strategy to reduce the electrical energy intake. This feature can be enforced at the design stage, by means of a lightweight design [5] or by a careful selection of the motor–reducer set [6,7] that overcomes the traditional approach based on inertia matching [8]. Another interesting and effective energy reduction technique is the exploitation of natural motion, which consists of tuning the motion characteristics with the elastic properties of the systems, so that it can be moved by relying minimally on the actuation effort [9]. This approach has been tested in several works [10,11,12,13] with good results, but similar results can be obtained by carefully designing an elastic element to reduce the energy consumption in a rest-to-rest motion [14,15].

The focus on motion properties as a relevant factor in the determination of the energy consumption is indeed not only limited to natural motion: a large body of studies has investigated the conditions that make a trajectory or a motion profile energy optimal. The alteration of the motion profile is often the least intrusive solution, as it generally does not require any physical alteration to the system [3], and, hence, it is a virtually ‘free’, but still effective, solution.

The generation of an energy-efficient motion profile is usually the result of some optimization technique based on a more or less detailed model of the system. One large group of studies focuses on the optimization performed on the basis of standard motion profiles [16], whose simple parametrization is well fitted to the determination of optimal motion conditions, as proven analytically in [17], or experimentally in [18]. The limitation to standard motion profiles, however, restricts the manifold in which the feasible solutions are located, given the usually very limited number of degrees of freedom that characterize the solution space. Some improvements can be found by designing a motion profile with a more efficient parametrization that is specifically tailored for the purpose, as in the work [19] that uses Chebyshev polynomials to build a rest-to-rest motion profile. Another option is the modulated cycloidal motion employed in [20] to generate energy-efficient motion profiles with null residual oscillation in one- and two-link flexible manipulators.

The two approaches just mentioned can, however, also be seen as direct approaches to an optimal control problem [21]. The generation of a motion profile can indeed be cast as a minimization problem constrained to some kinematic quantities (such as initial and final positions) and to the dynamics of the system [22]. If the cost function is the consumed energy, its minimization along a trajectory that connects the initial and final position/velocity within the respect of the system dynamics is precisely the ’true’ energy-optimal solution [23].

However, application of the aforementioned techniques has been limited in the context of underactuated robotic systems. Underactuated systems are ubiquitous in robotics and mechatronics [24], as many systems of wide practical use do belong to this class. Underactuation, i.e., the presence of less actuated degrees of freedom than the overall number of degrees of freedom, appears whenever some sort of flexibility is present in the system. This happens virtually in all flexible systems, such as in flexible link robots or in industrial robots, where some form of joint flexibility is often present. Overhead cranes [25] and underactuated cable-driven robots [26] do also belong to the class of underactuated systems, just to mention two common applications. The 2-DOFs RR planar manipulator used for the experimental test can therefore be used as a representative model to test novel control and planning techniques, due to its nonlinear dynamics and underactuation properties.

This work proposes the formulation of a constrained optimal control problem solution to generate a motion profile for underactuated systems that is characterized by minimum energy expenditure, that enforces arbitrary kinematic conditions at the boundaries and limited motor torque, and that results in theoretically null residual oscillation. The method is based on the development of an accurate nonlinear dynamic model, which accounts for the electromechanical dynamics of the manipulator under investigation. The latter is a 2-DOFs RR planar manipulator with just one actuated joint.

The contribution of this paper is to experimentally assess the benefits of an optimal control-based planner for generating minimum-energy point-to-point motion for underactuated systems, with the additional requirement of suppressing residual oscillations.

This work includes a description of the setup and its analytical dynamic model, then covers the method used to set up and solve the optimization problem. The resulting motion profiles are then tested experimentally on a custom lab setup and the results are compared, in terms of residual oscillations amplitude and in terms of absorbed electric energy, with some standard planning techniques based on input shaping.

2. Dynamic Model

The system under investigation is a planar two-link manipulator with revolute joints. A sketch of the studied manipulator is presented in Figure 1. The first joint is actuated by a motor, while the second joint is passive and equipped with a torsional spring that generates a restoring torque. The second revolute joint is also equipped with a rotary encoder that is, however, not used for the experiment. The system is underactuated, since it has two degrees of freedom and just one actuator.

The dynamics of the manipulator are described by the following equation (see for example [27])

$$\left[\begin{array}{c}{\tau }_1\\ 0\end{array}\right]=\mathbf{M}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}+\mathbf{g}\left(\mathbf{q}\right)+\mathbf{K}\mathbf{q}+{\mathbf{F}}_v\dot{\mathbf{q}}+{\mathbf{F}}_c\mathrm{sign}\left(\dot{\mathbf{q}}\right),$$

(1)

where $\mathbf{q}={[q_1,q_2]}^{\top }$ is the vector of joint angle coordinates, ${\tau }_1$ is the torque applied through the first joint, $\mathbf{M}\left(\mathbf{q}\right)$ is the (generalized) mass matrix, $\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})$ is the Coriolis matrix, $\mathbf{g}\left(\mathbf{q}\right)$ is the gravitational torque vector, $\mathbf{K}=\mathrm{diag}\left([0,k_s]\right)$ is the stiffness matrix with $k_s$ being the torsional spring constant, and ${\mathbf{F}}_v$ and ${\mathbf{F}}_c$ are the viscous and Coulomb friction coefficient matrices, respectively. The mass matrix is given by

$$\mathbf{M}\left(\mathbf{q}\right)=\left[\begin{array}{cc}{J}_1+J_2+J_m+(m_2+m_{\mathrm{enc}})a_1^2+2m_2{a}_1{\ell }_{2}cos{q}_2& J_2+m_2{a}_1{\ell }_{2}cos{q}_2\\ J_2+m_2{a}_1{\ell }_{2}cos{q}_2& J_2\end{array}\right],$$

(2)

where $m_1$, $m_2$ are the link masses, $m_{\mathrm{enc}}$ is the mass of the encoder and its mounting, $J_1$, $J_2$ are the link inertia referred to the joint axes, $J_m$ is the motor inertia, ${\ell }_1$, ${\ell }_2$ are the link center-of-mass positions, and $a_1$ is the length of the first link.

The Coriolis and centrifugal matrix is

$$\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})=\left[\begin{array}{cc}h{\dot{q}}_2& h({\dot{q}}_1+{\dot{q}}_2)\\ -h{\dot{q}}_1& 0\end{array}\right],$$

(3)

where $h=-m_2{a}_1{\ell }_{2}2sin\left(q_2\right)$. The gravitational torque vector is

$$\mathbf{g}\left(\mathbf{q}\right)=\left[\begin{array}{c}[m_1{\ell }_1+(m_2+m_{\mathrm{enc}})a_1]gcos{q}_1+m_{2}g{\ell }_{2}cos(q_1+q_2)\\ m_{2}g{\ell }_{2}cos(q_1+q_2)\end{array}\right],$$

(4)

where g is the gravitational acceleration. The forward dynamics can be derived by solving Equation (1) for $\ddot{\mathbf{q}}$:

$$\ddot{\mathbf{q}}={\mathbf{M}}^{-1}\left(\mathbf{q}\right)\left(\left[\begin{array}{c}{\tau }_1\\ 0\end{array}\right]-\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}-\mathbf{g}\left(\mathbf{q}\right)-\mathbf{K}\mathbf{q}-{\mathbf{F}}_v\dot{\mathbf{q}}-{\mathbf{F}}_c\mathrm{sign}\left(\dot{\mathbf{q}}\right)\right)={\mathbf{f}}_{\mathrm{D}}(\mathbf{q},\dot{\mathbf{q}},{\tau }_1),$$

(5)

where ${\mathbf{f}}_{\mathrm{D}}(·)$ denotes the forward dynamics map.

To enforce higher-order constraints, the time derivative of the acceleration (i.e., the jerk) is computed using the chain rule:

$$\stackrel{⃛}{\mathbf{q}}={\displaystyle \frac{\partial {\mathbf{f}}_{\mathrm{D}}}{\partial \mathbf{q}}}\dot{\mathbf{q}}+{\displaystyle \frac{\partial {\mathbf{f}}_{\mathrm{D}}}{\partial \dot{\mathbf{q}}}}\ddot{\mathbf{q}}+{\displaystyle \frac{\partial {\mathbf{f}}_{\mathrm{D}}}{\partial {\tau }_1}}{\dot{\tau }}_1={\mathbf{j}}_{\mathrm{D}}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}},{\tau }_1,{\dot{\tau }}_1),$$

(6)

where ${\mathbf{j}}_{\mathrm{D}}(·)$ denotes the jerk dynamics map.

The complete system dynamics can be written in state-space form. Define the state vector and control input as

$$\begin{array}{cc}\hfill \mathbf{x}& ={\left[\begin{array}{c}{\mathbf{q}}^{\top },{\dot{\mathbf{q}}}^{\top },{\ddot{\mathbf{q}}}^{\top },{\tau }_1\end{array}\right]}^{\top },\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill u& ={\dot{\tau }}_1.\hfill \end{array}$$

(8)

The state-space dynamics is then given by

$$\dot{\mathbf{x}}={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[\begin{array}{c}\mathbf{q}\\ \dot{\mathbf{q}}\\ \ddot{\mathbf{q}}\\ {\tau }_1\end{array}\right]=\left[\begin{array}{c}\dot{\mathbf{q}}\\ \ddot{\mathbf{q}}\\ {\mathbf{j}}_{\mathrm{D}}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}},{\tau }_1,u)\\ u\end{array}\right]=\mathbf{f}(\mathbf{x},u),$$

(9)

where $\mathbf{f}(\mathbf{x},u)$ denotes the overall state-space dynamics of the system.

Note that the map ${\mathbf{j}}_{\mathrm{D}}$ in Equation (9) can be substituted with any underactuated manipulator model. The proposed approach is intended to be general, and the 2-DOFs system is adopted here primarily for a practical experimental validation.

3. Minimum-Energy Planning

3.1. Requirement of the Trajectory

The goal of this work is to plan a point-to-point trajectory for the system presented in the previous section, but similar considerations are valid for many other underactuated systems. First of all, the initial and final configurations must be defined: they will serve as the kinematics boundary conditions. Since the system is underactuated, only one of the joint coordinates can be directly imposed. In this work, we choose to impose a value for $q_1$.

To determine the corresponding value of $q_2$, we require the system to be in static equilibrium, meaning both joint velocities and accelerations are zero. Given a specified value of $q_1$, this condition can be enforced by solving the following root-finding problem

$$0=g_2\left(\mathbf{q}\right)+k_s{q}_2=e_q\left(q_2\right),$$

(10)

which is derived by considering the second row of Equation (1) under the assumption $\dot{\mathbf{q}}=\ddot{\mathbf{q}}=\mathbf{0}$. Here, $g_2$ denotes the second component of the gravity vector $\mathbf{g}$ and $e_q$ is the equilibrium condition function. Once Equation (10) is solved for both the initial and final point, this results in the initial and final configurations ${\mathbf{q}}_0$ and ${\mathbf{q}}_f$.

In addition to reaching the desired configurations, the trajectory must satisfy specific boundary conditions to ensure stable motion. Specifically, the system should start and end at rest, meaning that the joint velocities must be zero at both the initial and final times. Moreover, zero acceleration is required at the final time to prevent residual oscillations and thus ensure that the system remains at the target configuration.

These requirements can be expressed mathematically as:

$$\left\{\begin{array}{c}\mathbf{q}\left(0\right)={\mathbf{q}}_0,\hfill \\ \mathbf{q}\left(t_f\right)={\mathbf{q}}_f,\hfill \\ \dot{\mathbf{q}}\left(0\right)=\mathbf{0},\hfill \\ \dot{\mathbf{q}}\left(t_f\right)=\mathbf{0},\hfill \\ \ddot{\mathbf{q}}\left(t_f\right)=\mathbf{0},\hfill \end{array}\right.\to \left\{\begin{array}{c}{\mathbf{A}}_0\mathbf{x}\left(0\right)={\mathbf{b}}_0,\hfill \\ {\mathbf{A}}_f\mathbf{x}\left(t_f\right)={\mathbf{b}}_f,\hfill \end{array}\right.$$

(11)

where $t_f$ is the final time, and ${\mathbf{A}}_0$, ${\mathbf{A}}_f$, ${\mathbf{b}}_0$, and ${\mathbf{b}}_f$ are appropriate matrices and vectors used to compactly express the boundary conditions in matrix form.

3.2. Objective Function

To plan a minimum-energy trajectory, we must define an appropriate objective function. In this system, only the first joint is actuated, and it is driven by a direct-drive DC motor. As a result, the motor torque is directly related to the armature current through

$$i_a={\displaystyle \frac{{\tau }_1}{k_t}},$$

(12)

where $i_a$ is the armature current and $k_t$ is the motor’s torque constant.

Using the standard electrical model of a DC motor, the armature voltage can be expressed as

$$v_a=R_a{i}_a+L_a{\displaystyle \frac{\mathrm{d}{i}_a}{\mathrm{d}t}}+k_v{\dot{q}}_1,$$

(13)

where $v_a$ is the armature voltage, $R_a$ is the armature resistance, $L_a$ is the armature inductance, and $k_v$ is the back-EMF constant. In this case, the inductance term has to be considered, as the system does not exhibit constant inertia [17].

The instantaneous electrical power consumed by the motor is then given by

$$\mathcal{P}=v_a{i}_a=R_a{i}_a^2+L_a{i}_a{\displaystyle \frac{\mathrm{d}{i}_a}{\mathrm{d}t}}+k_v{\dot{q}}_1{i}_a,$$

(14)

where $\mathcal{P}$ denotes the electrical power input to the motor.

Substituting Equation (12) into the expression above, and assuming $k_v=k_t$ (as we are considering DC motors), we obtain

$$\mathcal{P}={\displaystyle \frac{R_a}{k_t^2}}{\tau }_1^2+{\displaystyle \frac{L_a}{k_t^2}}{\tau }_1{\dot{\tau }}_1+{\dot{q}}_1{\tau }_1,$$

(15)

highlighting how the electrical power depends on the system states.

Finally, the total energy consumed by the motor during the task is given by

$$\mathcal{E}={\int }_0^{t_f}\mathcal{P} \mathrm{d}t,$$

(16)

where $t_f$ is the prescribed task duration and $\mathcal{E}$ represents the total energy consumption, which serves as the optimization objective.

3.3. Optimal Control Problem

This section formulates the minimum-energy trajectory planning task as an optimal control problem. Since the optimization variables are time-dependent functions rather than finite-dimensional vectors, the resulting formulation is infinite-dimensional, a class of problems commonly referred to as optimal control problems [28].

Using Equations (9), (11) and (16), the problem can be stated as

$$\begin{array}{cc}\hfill \underset{\mathbf{x}(·),u(·)}{minimize}& \mathcal{E}={\int }_0^{t_f}\mathcal{P}(\mathbf{x},u)\mathrm{d}t\hfill \end{array}$$

(17a)

$$\begin{array}{cc}\hfill \mathrm{subject}\mathrm{to}:& \dot{\mathbf{x}}\left(t\right)=\mathbf{f}(\mathbf{x}\left(t\right),u\left(t\right))\hfill \end{array}$$

(17b)

$$\begin{array}{cc}\hfill & {\mathbf{A}}_0\mathbf{x}\left(0\right)={\mathbf{b}}_0\hfill \end{array}$$

(17c)

$$\begin{array}{cc}\hfill & {\mathbf{A}}_f\mathbf{x}\left(t_f\right)={\mathbf{b}}_f\hfill \end{array}$$

(17d)

$$\begin{array}{cc}\hfill & \left|{\tau }_1\left(t\right)\right|\le {\tau }_{\lim }.\hfill \end{array}$$

(17e)

Here, ${\tau }_{\lim }$ denotes the limit torque that the motor can exert. Since the torque ${\tau }_1$ is part of the system state, the constraint in (17e) constitutes a state constraint. Consequently, the problem in Equation (17) is a nonlinear optimal control problem with state constraints. To solve such problems, numerical methods are typically employed. In particular, direct methods are preferred, as indirect methods often struggle to handle state constraints effectively [29].

The problem formulated in Equation (17) can be solved using standard direct-method-based solvers. In this work, the Rockit framework is employed for this purpose [30]. The solver is not in general real-time capable, as solving large nonlinear constrained optimization problems remains an open research problem.

4. Results

4.1. Experimental Setup

The experimental setup is composed by a two-link planar manipulator actuated only on the first joint by a “Speeder motion line—MB057DG218” motor. The second is passive and a torsional spring is made from two linear springs in parallel. The experimental setup is shown in Figure 2.

The system is controlled via Simulink Real-Time using a National Instruments PCIe-6321 data I/O board. The motor driver operates in current control mode.

Reference tracking is implemented through a standard position feedback architecture combining PID control with feedforward compensation, running at a control frequency of 1 kHz. The PID weights have been manually tuned to ensure minimal trajectory tracking error. The motion of the first and second joint coordinates is tracked using a Vicon 3D motion capture system, operating at a sampling frequency of 400 Hz. The electrical power consumed by the DC motor is computed as the product of the measured armature current and voltage. The total energy consumption is then obtained by numerical integration over time. The main physical parameters of the system are summarized in

Table 1. Data of the two-link underactuated serial manipulator studied in this work.

Parameter	Unit	Value
$m_1$	kg	$0.237$
$m_2$	kg	$0.021$
$m_{\mathrm{enc}}$	kg	$0.10$
$a_1$	m	$0.172$
$a_2$	m	$0.154$
${\ell }_1$	m	$0.086$
${\ell }_2$	m	$0.077$
$J_1$	kg m2	$3.601$$\times {10}^{-3}$
$J_m$	kg m2	$2.700\times {10}^{-5}$
$J_2$	kg m2	$1.660\times {10}^{-4}$
$f_{v,1}$	Nms/rad	$3.913\times {10}^{-3}$
$f_{v,2}$	Nms/rad	$1.949\times {10}^{-4}$
$f_{c,1}$	Nm	$3.431\times {10}^{-3}$
$f_{c,2}$	Nm	$6.455\times {10}^{-5}$
$k_s$	Nm/rad	$0.1949$
$R_a$	$\mathrm{\Omega }$	$1.7$
$L_a$	H	$3.39\times {10}^{-3}$
$k_t$	Nm/A	$0.071$

.

Two different test cases are studied to investigate the proposed optimal control based planner. Test case I consists of moving the the manipulator from the stable equilibrium point to a case where the ${\vartheta }_{1,f}=7\pi /4$, while test case II consists of a swing between symmetric points, i.e., ${\vartheta }_{1,0}=5\pi /4$ and ${\vartheta }_{1,f}=7\pi /4$. The data concerning the two test-cases are summarized in

Table 2. Test cases trajectories data.

Quantity	Unit	Test-Case I	Test-Case II
${\vartheta }_{1,0}$	rad	$3\pi /2$	$7\pi /4$
${\vartheta }_{1,f}$	rad	$5\pi /4$	$7\pi /4$
$t_f$	s	$1.0$	$1.5$

.

4.2. Test Case I

The first test case involves the execution of a rest-to-rest motion from the stable equilibrium position, i.e., from ${\vartheta }_1=3/2\pi$rad to ${\vartheta }_1=7/4\pi$rad. The performance of the energy-optimal motion profile designed according to the method outlined in Section 3 is assessed by a comparison with some shaped polynomial motion profile of degree 5. A fifth-order polynomial profile is chosen as it allows to simply enforce position, speed, and acceleration at the two boundaries. It must be pointed out that the execution of this motion by the first link of the robot would inevitably generate pronounced residual oscillations; therefore, the basic poly5 motion is filtered using input shaping. Input shaping consists of the alteration of a reference profile by convolving it with a sequence of a finite number of impulses with defined timing and amplitude, with the aim of achieving null residual oscillations. Input shapers are widely adopted for their simple implementation and their simple design, which requires just to measure the frequency and the damping factor of the oscillation mode to be suppressed [31,32].

The most basic and popular implementation of input shaping is the Zero Vibration (ZV) shaper, which achieves null residual vibration by placing the second pulse half of the damped oscillation period after the first one. A less common alternative is the Negative Zero Vibration (NZV) shaper, which allows for a reduced delay between consecutive pulses [33]. One drawback of the ZV and the ZVD shaper is their limited capability of coping with model uncertainties, so that any model-plant mismatch significantly affects the amount of residual oscillations. There are indeed some shaping solutions that are specifically designed to be more robust to model-plant mismatches, such as the Zero Vibration and Derivative (ZVD) shaper and the Negative Zero Vibration and Derivative (NZVD). These two solutions are quite convenient when model-plant mismatches are significant, as in the case of the uncertainty in the determination of oscillation modes or in the presence of significant model nonlinearities [34]. The latter is the case under consideration here, for which the determination of the modal properties is conducted by a linearization around the stable equilibrium point of the model of Equation (9).

The results are reported in Figure 3, Figure 4, Figure 5, Figure 6, and Figure 7, corresponding to the optimal trajectory, and the ZV-, ZVD-, NZV-, and NZVD-filtered trajectories, respectively.

From the energy point-of-view, it is worth noticing that the proposed trajectory allows a significant saving when compared to one of the other trajectories tested in test case I. In particular the evolution of the cumulative energy absorbed by the motor during the planned motion is depicted in Figure 8.

Furthermore, a quantitative comparison in terms of energy expenditure and residual oscillations is resumed in

Table 3. Test case I: comparison of energy consumption and residual oscillations.

Trajectory Type	Energy Consumption (J)	Oscillations (rad)
optimal	$8.044$	$3\times {10}^{-4}$
ZV	$9.161;(+13.89\%)$	$5\times {10}^{-4};(+66.67\%)$
ZVD	$10.048;(+24.91\%)$	$7\times {10}^{-4};(+133.33\%)$
NZV	$9.332;(+16.1\%)$	$10\times {10}^{-4};(+233.33\%)$
NZVD	$9.391;(+16.75\%)$	$7\times {10}^{-4};(+133.33\%)$

.

Notably, the proposed trajectory is comparable to the other methods in terms of peak residual oscillations, which remain quite limited in their order of magnitude. At the same time, using a non-optimized motion profile affects the energy expenditure quite significantly, more specifically in the range from $+13.89\%$ to $+16.75\%$. The data show, therefore, the improved performance brought by the proposed solution.

4.3. Test Case II

Test case II involves the motion between two symmetrical positions, in this case between ${\vartheta }_{1,0}=5\pi /4$ rad and ${\vartheta }_{1,f}=7\pi /4$ rad. As for the previous test case, the experimental results are provided by evaluating the position and speed of both links, as well as the electric energy intake, when executing the optimized motion and the shaped motion profiles as well. The results of the experimental trials are displayed in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 and in

Table 4. Test case II: comparison of energy consumption and residual oscillations.

Trajectory Type	Energy Consumption (J)	Peak Oscillation (rad)
optimal	$7.211$	$8.0\times {10}^{-3}$
ZV	$22.404;(+210.69\%)$	$2.5\times {10}^{-3};(-68.75\%)$
ZVD	$23.002;(+218.98\%)$	$3.0\times {10}^{-4};(-96.25\%)$
NZV	$22.363;(+210.12\%)$	$1.2\times {10}^{-3};(-85\%)$
NZVD	$22.802;(+216.21\%)$	$2.0\times {10}^{-4};(-75\%)$

.

From the energy point-of-view, it is worth noticing that the proposed trajectory allows a very significant saving, as all the others trajectories require at least 3 times more energy, as shown in terms of the evolution of the cumulative energy absorbed by the motor in Figure 14 and as summarized in Table 4.

Looking at the residual oscillations, however, the proposed optimal trajectory results in higher residual vibrations. This can be attributed to the significantly higher acceleration content of the proposed trajectory, which increases the sensitivity to unmodeled dynamics and is harder to track by the control device. Nevertheless, the residual oscillations remain an order of magnitude smaller than the total displacement and they are quickly damped, so they provide a minimal detrimental effect to the quality of the overall motion.

Compared to the ZV filter (another non-robust method), the proposed trajectory achieves residual vibrations below $2\times {10}^{-3}$ rad in $0.5$ s, whereas the ZV filter reaches the same level in $0.4$ s. The remaining filters suppress the oscillations almost instantaneously.

5. Conclusions

This work presents an optimal-control-based planner for generating energy-efficient trajectories for underactuated manipulators. The formulation accommodates torque constraints and explicitly enforces oscillation suppression at the final time. The trajectory optimization problem is solved using an open-source solver.

The approach has been validated experimentally on a planar two-degree-of-freedom manipulator with a passive second joint. Experimental results demonstrate significant energy savings while maintaining low residual oscillations.

The experimental validation demonstrates robustness to model mismatches—such as non-ideal friction and nonlinear stiffness—highlighting the potential of the approach for deployment in more complex and realistic scenarios.

The method has been benchmarked against common input shaping techniques (ZV, ZVD, NZV, and NZVD), resulting in comparable residual oscillations. Future work will focus on extending the proposed approach to other robotic systems.