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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Robot motor capability is a crucial factor for a robot, because it affects how accurately and rapidly a robot can perform a motion to accomplish a task constrained by spatial and temporal conditions. In this paper, we propose and derive a pseudo-index of motor performance (_{p}_{p}_{p}

Robots are widely-used in manufacturing tasks, such as assembly [

Robot motor capability is the motor ability for a robot to accomplish a task constrained by spatial and temporal conditions. Traditionally, a robot was evaluated by its accuracy and repeatability [

In this paper, the proposed method for characterizing the motor capability of a robot is inspired by Fitts's law, which is one of the well-known metrics for studying human rapid movements. Fitts's law was revealed by Paul Fitts in 1954, showing the information capacity of a human motor system. In Fitts's law, the motor performance is the ability to consistently produce movements and is described by two main factors—speed and accuracy. Fitts obtained the following equation, often called Fitts's law, as:
_{mt}_{d}_{p}_{p}

Actually, Fitts's equation is a special case of the quasi-power function with a varying exponent, _{p}_{1} Thus,
_{1} results in smaller _{mt}

The paper proposes the method to derive the speed-accuracy constraint of a robot system the same as _{p}_{p}_{p}

This paper is organized as follows. In Section II, we formulate a robot task as an optimization problem constrained by the proposed speed-accuracy constraint derived from the kinematics, dynamics and control of a robot motor system. In Section III, we present the proposed pseudo-index of motor performance (_{p}_{p}

We formulate a robot task as an optimization problem subject to spatial and temporal constraints. The proposed speed-accuracy constraint characterizes the relationship between the robot joint speed and the Cartesian position error of the end-effector of the robot. By utilizing the proposed speed-accuracy constraint, the optimization problem is solved without violating the robot motor capability. We first formulate a basic rapid robot movement that moves its end-effector to reach a desired joint location, _{1}, _{2}_{i}_{n}^{T}_{x}, ε_{y}_{z}_{1}, _{2}, …, _{i}_{n}^{T}_{2} is the Euclidean-norm operation, _{d}_{x}, d_{y}_{z}_{x}, ε_{y}_{z}^{max}_{x}, ε_{y}_{z}

Since the spatial inaccuracy of the end-effector of a robot with revolute joints is caused by the inaccuracy of robot kinematics and dynamics models and the disturbance of the environment [_{x}, d_{y}_{z}_{d}

To derive Δ

In _{ij}_{i}, gravity terms, _{i}_{i}^{i}_{i}

In order not to excite the mechanical resonant frequency of the manipulator, the undamped natural frequency is set to no more than one-half of the structure resonant frequency. Complying to this constraint, we can obtain the following relation and the upper bound of
_{i}_{i}_{ii}_{i}

To determine Δ_{d}_{i}_{ijk}

Replacing the elements of Δ_{i}_{1}_{1}_{N}_{N}_{1}(_{N}^{T}_{1},…, Δ_{N}^{T}^{2} x ^{2} matrix, and Δ_{1}(_{n}^{2} x ^{2} x 1 matrix and

From _{new}_{old}, κ_{x}_{y}_{z}|) will be generated. Thus,

The purpose of determining the pseudo-index of the performance, _{p}_{p}_{p}_{d}_{d} = _{0}, where _{px}, pI_{py}_{pz}_{x}_{y}_{z}_{0} is an N^{2} x N matrix and
_{0} is an N^{2} x 1 matrix and

From _{x}, d_{y}, d_{z}_{x}_{x}^{T}_{d}_{p}_{d}_{x}_{p}

Computer simulations were performed by MATLAB Toolbox [_{p}

_{mt}

In this simulation, a PUMA 560 industrial robot was asked to move with the maximum speed to hit the target position within the spatial tolerances, _{x}_{y}_{d}_{x}_{y}

First, we evaluated how the Cartesian position error was affected by Δ_{diff}_{mt}_{diff}/T_{mt}_{diff}_{mt}_{d}

When the errors between the computed and actual masses,

Δ_{x}_{y}_{z}

Δ_{x}_{y}

Δ_{i}^{2}, because _{d}^{T}^{2}) (^{2}) when Δ_{i}

Δ

Then, we evaluated the _{p}^{c}, m^{c}_{mt}_{d}_{0} is the nominal joint velocities finishing
_{mt}_{d}

_{x}_{mt}_{px}_{px}

To verify the robot had different motor capabilities with respect to _{px}

To validate the speed-accuracy constraint, we performed the experiment on a PUMA 560 robot (see _{diff}_{mt}_{diff}/T_{mt}_{0}(rad/sec), where _{0} = _{diff}/_{diff}_{mt}_{2} = 1.5_{1}, _{3}_{1} and _{4} = _{1}. Although we could not obtain the actual masses of the links, we performed various Δ_{x},d_{y}

From _{2}_{3} and _{4} to the Cartesian position errors caused by _{1} in ^{2} ((1.5)^{2} = 2.25, (2)^{2} = 4, (2.5)^{2} = 6.25), since _{2} = 1.5_{1}, _{3} = _{1} and _{4} = _{1}. This shows that the Cartesian position error, ∝ ^{2}, is under the same Δ_{px}

To demonstrate that the pseudo-index of performance (_{px}_{px}_{px}

The proposed speed-accuracy constraints showed the relationship between Cartesian position errors of the end-effector of a robot and joint velocities. This relationship can be illustrated by a vector representation, as shown in _{a}_{ll}, is composed of _{x}_{y}_{V}_{G}_{S}, where Δ_{V}_{G}_{S}_{G}_{G}_{m}_{m}m_{m}_{S}_{S}_{m}_{s}^{2}, where _{s}_{0},_{0} is a reference speed and _{s}_{s}_{all}

From

The pseudo-index of performance is also affected by the target joint positions, _{d}_{x}, d_{y}_{z}^{1}-norm of
_{x}_{y}_{z}^{1}-norm indicates the maximum Cartesian position errors, _{x}, d_{y} and _{z}_{d}_{(}_{j}_{1)}:_{(}_{j2}_{)}:_{(}_{j3}_{)}:_{(}_{j4}_{)}:_{(}_{j5}_{)}:_{(}_{j6}_{)}, where _{(}_{ji}_{)} is a sampling index by joint resolution within the joint range of joint, _{(}_{j}_{1)} are the most significant bits (MSB) and _{(}_{j6}_{)} are the least significant bits (LSB). The index number represents the joint positions for all joints. To illustrate the concept and simplify the computation, joints 4, 5 and 6 are fixed as zero-degree, and the index number is only generated from joints 1, 2 and 3 with 20-degree joint resolution. ^{1}-norm on _{x}, d_{y}_{z}^{1}-norm on _{x}^{1}-norm on _{y}^{1}-norm on _{x}_{x}_{y}

In this paper, we have developed and presented a quantitative measure, the pseudo-index of performance (_{p}_{p}_{p}_{p}

The authors declare no conflict of interest.

Proportional-plus-derivative (PD) control scheme of a robot joint,

Quadratic function of the Cartesian position error of the end-effector of a robot along the _{x}_{px}^{-}^{1} and _{d}_{0}.

Scheme of our task, where the moving position error is defined as the error between the actual and target positions. (S: start position, G: target position and E: end position).

Simulation results of the Cartesian position errors of the end-effector of a PUMA 560 robot. (a) Case 1: Δ

Simulation results of the Cartesian position errors of the end-effector of a PUMA 560 robot in Case 3 when the joint velocities are non-zero: Δ

Simulation results of the Cartesian position errors of the end-effector of a PUMA 560 robot along the x-axis (_{x}_{mt}

Simulation results of the Cartesian position errors of the end-effector of a PUMA 560 robot along the x-axis (_{x}

PUMA 560 robot in the experiment.

Experimental results of the Cartesian position errors of the end-effector caused by the changes of joint velocities (

Experimental results of the Cartesian position errors of the end-effector caused by the fixed Am and the changes of the joint velocities (

PUMA 560 experimental results of the Cartesian position errors of the end-effector along the _{x}_{mt}

PUMA 260 experimental results of the Cartesian position errors of the end-effector along the _{x}_{mt}

Vector representation of Cartesian position errors.

Cartesian position errors with respect to index number. Joint resolution is 20 degrees. (_{x}_{y}_{z}^{1}-norm values on _{x}_{y}

Four movement velocities, D_{mt}

_{mt} |
||||
---|---|---|---|---|

0.0508 | 0.0508/0.392 | 0.0508/0.281 | 0.0508/0.212 | 0.0508/0.180 |

0.1016 | 0.1016/0.484 | 0.1016/0.372 | 0.1016/0.260 | 0.1016/0.203 |

0.2032 | 0.2032/0.580 | 0.2032/0.469 | 0.2032/0.357 | 0.2032/0.279 |

0.4064 | 0.4064/0.731 | 0.4064/0.595 | 0.4064/0.481 | 0.4064/0.388 |

Ratios of the Cartesian position errors of the end-effector of the PUMA 560 robot caused by _{2},_{3} and _{4} to the Cartesian position errors caused by _{1}.

_{2} |
_{3} |
_{4} | |
---|---|---|---|

^{2} = 2.25 |
^{2} = 4 |
^{2} = 6.25 | |

−3% | 2.2115 | 4.1868 | 6.3067 |

−2% | 2.1633 | 3.9134 | 6.3435 |

−1% | 2.2556 | 4.1400 | 6.1096 |

0% | 2.1339 | 4.0218 | 6.2426 |

1% | 2.1253 | 4.1400 | 6.1096 |

2% | 2.1066 | 3.8078 | 6.2284 |