2.1. Inverse kinematics (IK) for redundant manipulators

The main methods for solving the inverse kinematics problem of redundant robots can be broadly classified into three categories: joint-velocity-based methods, joint-position-based methods, and AI-based methods [Reference Hassan, El-Habrouk and Deghedie23]. Among them, the most representative velocity-based method is the gradient projection method. Colomé [Reference Colomé and Torras24] provided a detailed overview of research on the gradient projection method, such as minimizing joint-torque energy [Reference Zhao, Yang, Zhao and Liu25] and enhancing dexterity [Reference Patel and Sobh26], including joint-torque energy minimization and dexterity enhancement. JongHo Kim [Reference Kim, Shin and Jang27] reformulated redundant inverse kinematics as a gradient-based constrained optimization problem, numerically solved using sequential quadratic programming. AI-based methods establish the pose-to-joint mapping relationship through offline training, which can effectively avoid the computational burden of real-time calculation. However, their performance is limited by sample dependency and model generalization capability.

2.2. Decomposition of oriented maximum velocity

In serial robots, the relationship between the robot’s joint velocity and workspace velocity can be expressed as:

(1) \begin{equation}{\, \dot{\kern-2pt X}}=\left[\begin{array}{c} {\, \dot{\kern-2pt x}}_{T}\\[2pt] {\, \dot{\kern-2pt x}}_{R} \end{array}\right]=\left[\begin{array}{c} J_{T}\\[2pt] J_{R} \end{array}\right]\dot{\theta }=J\dot{\theta }\end{equation}

where $\dot{\theta }\in \mathbb{R}^{n\times 1}$ is the joint velocity vector, n is the number of joints, and ${\, \dot{\kern-2pt X}}\in \mathbb{R}^{m\times 1}$ is an m-dimensional column vector composed of the translational velocity ${\, \dot{\kern-2pt x}}_{T}$ and the rotational velocity ${\, \dot{\kern-2pt x}}_{R}$ of the robot’s end effector. The value of m depends on the degrees of freedom of the robot’s end-effector workspace. $J\in \mathbb{R}^{m\times n}$ is the Jacobian matrix composed of the translational Jacobian $J_{T}$ and the rotational Jacobian $J_{R}$ . Without loss of generality, we assume that the robot’s end-effector workspace is 6-dimensional, i.e., m = 6, in the following discussions.

The robot’s joint velocity vector is constrained by the maximum velocity of each joint, expressed by the infinity norm:

(2) \begin{equation}\left\|{\dot{\theta}}\right\|_{\infty}\le 1\end{equation}

The joint velocity vector of the robot can be described as the sum of the joint velocity vector that generates only translational velocity $\dot{\theta }_{T}$ in the workspace and the joint velocity vector that generates only rotational velocity $\dot{\theta }_{R}$ in the workspace, as shown in

(3) \begin{equation}\dot{\theta }=\dot{\theta }_{T}+\dot{\theta }_{R}\end{equation}

The translational-only joint velocity vector satisfies:

(4) \begin{equation}J_{R}\dot{\theta }_{T}=0\end{equation}

Similarly, the rotational-only joint velocity vector satisfies:

(5) \begin{equation}J_{T}\dot{\theta }_{R}=0\end{equation}

2.3. Solving maximum velocity using the DTF method

After the velocity decomposition in Eq. (3), the DTF method is the current method used to solve for the maximum velocity. This method maps the end-effector velocity to the joint velocity using the generalized inverse matrix and expresses the joint velocity vector under the constraints of each joint’s maximum velocity in the form of the infinity norm.

The translation velocity $\dot{\hat{x}}_{T}$ under the condition of zero rotational velocity is:

(6) \begin{equation}\dot{\hat{x}}_{T}=J_{T}\big(I_{n}-{J}_{R}^{^{+} }J_{R}\big)\,\dot{\theta }={\, \tilde{ J}}_{T}\dot{\theta }\end{equation}

where ${J}_{R}^{+}\in \mathbb{R}^{n\times 3}$ is the pseudo Moore–Penrose inverse of the rotational Jacobian matrix $J_{R}$ , and ${\, \tilde{\kern-2pt J}}_{T}$ represents the augmented translational Jacobian matrix under the zero rotational velocity constraint, $I_{n}$ is the n-dimensional identity matrix.

Similarly, the rotational velocity $\dot{\hat{x}}_{R}$ under the condition of zero translational velocity is:

(7) \begin{equation}\dot{\hat{x}}_{R}=J_{R}\big(I_{n}-{J}_{T}^{^{+} }J_{T}\big)\,\dot{\theta }={\, \tilde{ J}}_{R}\dot{\theta }\end{equation}

where ${J}_{T}^{+}\in \mathbb{R}^{n\times 3}$ is the pseudo Moore–Penrose inverse of the translational Jacobian matrix $J_{T}$ , and ${\,\, \tilde{\kern-2pt J}}_{R}$ represents the augmented rotational Jacobian matrix under the zero translational velocity constraint.

From (6) and (7), (3) can be transformed into:

(8) \begin{equation}\dot{\theta }=\dot{\theta }_{T}+\dot{\theta }_{R}={{\, \tilde{\kern-2pt J}}}_{T}^{^{+} }{\, \dot{\kern-2pt x}}_{T}+{{\, \tilde{\kern-2pt J}}}_{R}^{^{+} }{\, \dot{\kern-2pt x}}_{R}\end{equation}

where ${{\, \tilde{\kern-2pt J}}}_{T}^{+}$ is the pseudo Moore–Penrose inverse of the augmented translational Jacobian matrix ${\, \tilde{\kern-2pt J}}_{T}$ considering the constraint of zero rotational velocity, and ${{\, \tilde{\kern-2pt J}}}_{R}^{+}$ is the pseudo Moore–Penrose inverse of the augmented rotational Jacobian matrix ${\, \tilde{\kern-2pt J}}_{R}$ considering the constraint of zero translational velocity.

For a redundant manipulator, the solution consists of a particular solution plus a null-space component:

(9) \begin{equation}\dot{\theta }=J^{^{+} }{\, \dot{\kern-2pt x}}+\big(I_{n}-J^{^{+} }J\big)z=J^{^{+} }{\, \dot{\kern-2pt x}}+n\end{equation}

The joint velocities that produce only end-effector translational motion $\dot{\theta }_{T}$ and only end-effector rotational motion $\dot{\theta }_{R}$ are expressed as follows:

(10) \begin{align}\dot{\theta }_{T}={{\, \tilde{\kern-2pt J}}}_{T}^{^{+} }{\, \dot{\kern-2pt x}}_{T}+\alpha n \\[-25pt] \nonumber \end{align}

(11) \begin{equation}\dot{\theta }_{R}={{\, \tilde{\kern-2pt J}}}_{R}^{^{+} }{\, \dot{\kern-2pt x}}_{R}+\beta n\end{equation}

Here, ${\widetilde{J_{T}}^{+}}{\, \dot{\kern-2pt x}}_{T}$ and ${\widetilde{J_{R}}^{+}}{\, \dot{\kern-2pt x}}_{R}$ represent the particular solutions for pure translation and pure rotation, respectively. $n\in \text{Null}(J)$ denotes the null-space solution vector, and $\alpha$ and $\beta$ represent the contributions of the null-space vectors in the translational and rotational directions, respectively. The sum of these contributions is constrained by the joint velocity limits (Eq. 2)

Therefore, Eq. (8) in the case of redundancy is expressed as follows:

(12) \begin{equation}\dot{\theta }=\big(\dot{\theta }_{T}+\dot{\theta }_{R}\big)={{\, \tilde{\kern-2pt J}}}_{T}^{^{+} }{\, \dot{\kern-2pt x}}_{T}+{{\, \tilde{\kern-2pt J}}}_{R}^{^{+} }{\, \dot{\kern-2pt x}}_{R}+\left(\alpha +\beta \right)\!n\end{equation}

Because the DTF method does not address the null-space issue under redundancy, in our subsequent experiments—when computing in MATLAB—the solver effectively considered only the particular solutions, i.e., ${\,\hat{\kern-2pt J}}_{T}^{+}\dot{\mathbf{x}}_{T}$ and ${\,\hat{\kern-2pt J}}_{R}^{+}\dot{\mathbf{x}}_{R}$ . For the null-space components in (10) and (11), it implicitly set the null-space term to zero during the solution process. This approach yields the minimum $\ell_{2}$ -norm solution; it only guarantees, for the current configuration, minimal effort (or joint activity) when moving along the prescribed target direction, and does not necessarily maximize the velocity along that direction.

After discussing the handling of the DTF method in both nonredundant and redundant cases, it is clear that DTF is only applicable to nonredundant cases. Therefore, the following discussion will focus on the nonredundant case.

Let $h\in \mathbb{R}$ denote the relative ratio between the magnitudes of the end-effector’s translational and rotational velocities:

(13) \begin{equation}\tilde{h}=\tilde{v}/\tilde{\omega }\end{equation}

Then, the translation velocity vector $\dot{\hat{x}}_{T}$ and the rotational velocity vector $\dot{\hat{x}}_{R}$ of the end-effector can be expressed as:

(14) \begin{align}{\, \dot{\kern-2pt x}}_{T}=\parallel\! {\, \dot{\kern-2pt x}}_{T}\!\parallel _{2}\,u_{T}=\tilde{v}\,u_{T} \\[-25pt] \nonumber \end{align}

(15) \begin{equation}{\, \dot{\kern-2pt x}}_{R}=\parallel\! {\, \dot{\kern-2pt x}}_{R}\!\parallel _{2}\,u_{R}=\tilde{\omega }\,u_{R}\end{equation}

where $\tilde{v}$ is the magnitude of the translational velocity and $\tilde{\omega }$ is the magnitude of the rotational velocity, $u_{T},u_{R}\in \mathbb{R}^{3\times 1}$ are the directional vectors of translational velocity and rotational velocity in the workspace, respectively.

Substituting Eqs. (8), (20), (21), (22) into the constraints of Eq. (2) yields:

(16) \begin{equation}\left\| \tilde{v}\,J_T^{+}u_T + \tilde{\omega}\,J_R^{+}u_R \right\|_{\infty} \leq 1\end{equation}

Thus, the maximum translational velocity $\tilde{v}_{\max }$ and the maximum rotational velocity $\tilde{\omega }_{\max }$ can be expressed as:

(17) \begin{align} \tilde{v}_{\max} &= \big\| {\, \tilde{\kern-2pt J}}_{T}^+ u_T + h^{-1} {\, \tilde{\kern-2pt J}}_{R}^+ u_R \big\| _{\infty}^{-1} \nonumber \\[2pt] \tilde{\omega}_{\max} &= \big\|h \,{\, \tilde{\kern-2pt J}}_{T}^+ u_T + {\, \tilde{\kern-2pt J}}_{R}^+ u_R \big\|_{\infty}^{-1} \\[-10pt] \nonumber \end{align}

Therefore, given the motion parameters $(u_{T}, u_{R}, h)$ and joint velocity constraints, the maximum velocity in the specified direction at a target configuration can be approximately determined.

However, the DTF method has the following limitations:

• • The DTF method computes the $\ell_{2}$ -norm solution using the Moore–Penrose pseudoinverse, which generally corresponds to the minimum joint effort (or minimum joint activity) when moving along the prescribed target direction. However, in redundant cases, this criterion does not match the objective of maximizing the end-effector velocity.

• • When joint limits are unequal, it is unclear which joint dominates the infinity norm, complicating constraint handling.

3.1. Algorithmic procedure and flowchart

As shown in Figure 1, we take the pure translational motion of the robot as an example to illustrate the detailed implementation procedure of the proposed algorithm. To clearly present the theoretical basis and solution procedure of the proposed method, this section is organized into three parts. First, in 3.2, the joint velocity decomposition formulations for nonredundant and redundant robots are derived, where the joint velocity is decoupled into the component responsible for pure translational motion of the end-effector and the component responsible for pure rotational motion. Then, in 3.3, the joint velocity is mapped into the end-effector workspace through the velocity Jacobian matrix, and this mapping relationship is further expressed as a function of the unit vector of the specified motion direction, the ratio between translational and rotational motions, and the maximum velocity. Finally, in 3.4, the constructed function is further converted into a set of linear programming (LP) equations, thereby enabling the exact calculation of the maximum velocity along the specified direction

Figure 1.

Algorithm flowchart.

3.2. Joint velocity analysis

The null-space solution of the Jacobian matrix $J$ can be expressed as:

(18) \begin{equation}n=\sum\limits_{i=1}^{n-r\left(J\right)}t_{i}\,\sigma _{i}\end{equation}

The solution vector $\dot{\theta }_{T}$ that satisfies Eq. (10) can be expressed in terms of the fundamental solution set of the system of equations:

(19) \begin{equation}\dot{\theta }_{T}={{\, \tilde{\kern-2pt J}}}_{T}^{^{+} }{\, \dot{\kern-2pt x}}_{T}+\alpha \sum\limits_{i=1}^{n-r\left(J\right)}t_{i}\,\sigma _{i}\end{equation}

where $\sigma _{i}$ represents the i-th column of the null space matrix $N(J)$ of the Jacobian matrix $J$ , with $N(J)\in \mathbb{R}^{n\times (n-r(J))}$ , and $r(J)$ denoting the rank of $J$ . $t_{i}$ is an arbitrary constant, where $i=1,2,\cdots ,n-r(J).$

Similarly, the joint velocity vector generating only rotational velocity is:

(20) \begin{equation}\dot{\theta }_{R}={{\, \tilde{\kern-2pt J}}}_{R}^{^{+} }{\, \dot{\kern-2pt x}}_{R}+\beta \sum\limits_{i=1}^{n-r\left(J\right)}t_{i}\,\sigma _{i}\end{equation}

From Eq. (12), $\dot{\theta }$ can be simplified as a function of the unknowns $t_{i}$ , $\alpha$ , and $\beta$ , as follows:

(21) \begin{equation}\dot{\theta }\!\left(t_{i},\alpha ,\beta \right)={{\, \tilde{\kern-2pt J}}}_{T}^{^{+} }{\, \dot{\kern-2pt x}}_{T}+{{\, \tilde{\kern-2pt J}}}_{R}^{^{+} }{\, \dot{\kern-2pt x}}_{R}+\left(\alpha +\beta \right)\sum\limits_{i=1}^{n-r\left(J\right)}t_{i}\,\sigma _{i}\end{equation}

Without loss of generality, we take the case where the maximum joint velocity for each joint is set to ±1 rad/s as an example. This constraint can be obtained by substituting Eq. (22) into Eq. (2), as follows:

(22) \begin{equation}\Bigg\| {{\, \tilde{\kern-2pt J}}}_{T}^{^{+} }{\, \dot{\kern-2pt x}}_{T}+{{\, \tilde{\kern-2pt J}}}_{R}^{^{+} }{\, \dot{\kern-2pt x}}_{R}+\left(\alpha +\beta \right)\sum\limits_{i=1}^{n-r\left(J\right)}t_{i}\,\sigma _{i}\Bigg\| _{\mathrm{\infty }}\leq 1\end{equation}

In fact, different joint velocity limits can be adjusted in Eq. (35).

3.3. Joint velocity analysis velocity analysis in workspace

The robot’s end-effector velocity vector ${\, \dot{\kern-2pt X}}\in \mathbb{R}^{m\times 1}$ can be decomposed into the translational velocity vector ${\, \dot{\kern-2pt x}}_{T}$ and the rotational velocity vector ${\, \dot{\kern-2pt x}}_{R}$ . Then, ${\, \dot{\kern-2pt x}}_{T}$ and ${\, \dot{\kern-2pt x}}_{R}$ can be expressed as the product of a constant $k_{T}$ and $k_{R}$ , and the corresponding unit directional vector for translation ${\, \dot{\kern-2pt x}}_{\textit{unitT}}$ and rotation ${\, \dot{\kern-2pt x}}_{\textit{unitR}}$ , respectively.

(23) \begin{equation}{\, \dot{\kern-2pt X}}=\left[\begin{array}{c} {\, \dot{\kern-2pt x}}_{T}\\[2pt] {\, \dot{\kern-2pt x}}_{R} \end{array}\right]=\left[\begin{array}{c} k_{T}{\, \dot{\kern-2pt x}}_{\text{unit}T}\\[2pt] k_{R}{\, \dot{\kern-2pt x}}_{\text{unit}R} \end{array}\right]\end{equation}

Let $d\in \mathbb{R}$ denote the relative ratio between the magnitudes of the end-effector’s translational and rotational velocities:

(24) \begin{equation}d=\frac{k_{T}}{k_{R}}\end{equation}

From Eqs. (1) and (3):

(25) \begin{equation}{\, \dot{\kern-2pt X}}=k_{R}\bigg[\begin{array}{c} d\cdot {\, \dot{\kern-2pt x}}_{\text{unit}T}\\[3pt] {\, \dot{\kern-2pt x}}_{\text{unit}R} \end{array}\bigg]=J\dot{\theta }=J\left(\dot{\theta }_{T}+\dot{\theta }_{R}\right)\end{equation}

Substituting (22) into (26):

(26) \begin{equation}k_{R}\bigg[\begin{array}{c} d\cdot {\, \dot{\kern-2pt x}}_{\text{unit}T}\\[3pt] {\, \dot{\kern-2pt x}}_{\text{unit}R} \end{array}\bigg]=J\left({{\, \tilde{\kern-2pt J}}}_{T}^{^{+} }{\, \dot{\kern-2pt x}}_{T}+{{\, \tilde{\kern-2pt J}}}_{R}^{^{+} }{\, \dot{\kern-2pt x}}_{R}+\left(\alpha +\beta \right)\sum\limits_{i=1}^{n-r\left(J\right)}t_{i}\,\sigma _{i}\right)\end{equation}

Define:

(27) \begin{equation}f\left(t_{i},\alpha ,\beta \right)=J\left({{\, \tilde{\kern-2pt J}}_{T}}^{+}{\, \dot{\kern-2pt x}}_{T}+{{\, \tilde{\kern-2pt J}}_{R}}^{+}{\, \dot{\kern-2pt x}}_{R}+\left(\alpha +\beta \right)\sum\limits_{i=1}^{n-r(J)}t_{i}\,\sigma _{i}\right)\end{equation}

Let ${\, \dot{\kern-2pt x}}_{new}$ represent $\left[\begin{array}{c} d\cdot {\, \dot{\kern-2pt x}}_{\text{unit}T}\\ {\, \dot{\kern-2pt x}}_{\text{unit}R} \end{array} \right]$ . Since ${\, \dot{\kern-2pt x}}_{new}$ is an m-dimensional vector, Eq. (28) can be regarded as a system of m equations. Substituting Eq. (28) into (27), the system of equations can be expressed as:

(28) \begin{align}\begin{array}{c} k_{R}\cdot {\, \dot{\kern-2pt x}}_{\mathrm{new}g}=f_{g}\left(t_{i},\alpha ,\beta \right)\\[3pt] g=1,2,\ldots ,m\\[3pt] i=1,2,\ldots ,n-r\left(J\right) \end{array} \end{align}

where g denotes the g-th equation in Eq. (28), ${\, \dot{\kern-2pt x}}_{newg}$ is the g-th element in the column vector ${\, \dot{\kern-2pt x}}_{new}$ and $f_{g}(t_{i},\alpha ,\beta )$ represent the g-th row of $f (t_{i},\alpha ,\beta )$ . Equation (29) exhibits the relationship between the velocity coefficient $k_{R}$ and the variables $t_{i}$ , $\alpha$ , and $\beta$ . Therefore, the maximum velocity of the robot along the specified coupled translational and rotational motion direction $\,{\dot{\kern-2pt X}}$ is $k_{{R_{{\max }}}}{\, \dot{\kern-2pt x}}_{{new}}$ , where $k_{{R_{{\max }}}}$ is the maximum value that $k_{R}$ can take under the constraint conditions specified in Eq. (27).

3.4. Conversion into linear programming equations

Transform the first equation containing $k_{R}$ in Eq. (29) into an objective function of $k_{R}$ in terms of $t_{i}$ . Assuming the selected equation is the first equation, we have

(29) \begin{align}\begin{array}{c} k_{R}=F\!\left(t_{i},\alpha ,\beta \right)=\dfrac{f_{1}\!\left(t_{i},\alpha ,\beta \right)}{{\, \dot{\kern-2pt x}}_{\mathrm{new}1}}\\[8pt] i=1,2,\ldots ,n-r\left(J\right) \end{array}\end{align}

By eliminating $k_{R}$ from the m equations in Eq. (34), m-1 equations without $k_{R}$ can be obtained. These m-1 equations can be used as equality constraints and can be expressed as:

(30) \begin{equation}\mathrm{s}.\mathrm{t}.\quad \begin{cases} \dfrac{f_{1}\!\left(t_{i},\alpha ,\beta \right)}{{\, \dot{\kern-2pt x}}_{\mathrm{new}1}}{\, \dot{\kern-2pt x}}_{\mathrm{new}2}-f_{2}\!\left(t_{i},\alpha ,\beta \right)=0,\\[8pt] \dfrac{f_{1}\!\left(t_{i},\alpha ,\beta \right)}{{\, \dot{\kern-2pt x}}_{\mathrm{new}1}}{\, \dot{\kern-2pt x}}_{\mathrm{new}3}-f_{3}\!\left(t_{i},\alpha ,\beta \right)=0,\\[4pt] \vdots \\[4pt] \dfrac{f_{1}\!\left(t_{i},\alpha ,\beta \right)}{{\, \dot{\kern-2pt x}}_{new1}}{\, \dot{\kern-2pt x}}_{{new_{m}}}-f_{m}\!\left(t_{i},\alpha ,\beta \right)=0. \end{cases}\end{equation}

For simplicity of expression, we represent the above equation in the following form:

(31) \begin{align}\begin{array}{c} y_{p}\!\left(t_{i},\alpha ,\beta \right)=0\\[3pt] p=1,2,\ldots ,m-1\\[2pt] i=1,2,\ldots ,n-r\!\left(J\right) \end{array}\end{align}

where $y_{p}(t_{i},\alpha ,\beta )$ represents the p-th of the m-1 equations after eliminating $k_{R}$ , and the Eq. (17) can be transformed into n inequality constraints, as follows:

(32) \begin{equation}-1\leq \dot{\theta }_{q}\!\left(t_{i},\alpha ,\beta \right)\leq 1,\quad q=1,2,\ldots ,n\end{equation}

It should be noted that in Eq. (23), we use a joint velocity limit of ±1 rad/s as an example, resulting in the Eq. (33). In fact, the joint velocity limits at the corresponding joints can be adjusted by modifying the velocity range of any row in Eq. (27). $\dot{\theta }_{q}(t_{i},\alpha ,\beta )$ represents the q-th row of $\dot{\theta }(t_{i},\alpha ,\beta )$ in Eq. (22).

In the case of heterogeneous joint-velocity limits:

(33) \begin{equation}-\dot{\theta }_{q\max }\leq \dot{\theta }_{q}\leq \dot{\theta }_{q\max }\end{equation}

Here, $\dot{\theta }_{qmax}$ denotes the maximum (limit) velocity of the $q$ -th joint.

When a component of $\dot{\theta }$ reaches its joint-velocity limit, the robot’s speed along $\dot{\boldsymbol{x}}$ attains its maximum as well. Accordingly, the maximum speed scaling factor $k_{Rmax}$ is computed as follows:

(34) \begin{equation}k_{R\max }=\min _{i}\left\{\left |\frac{\dot{\theta }_{q\max }}{\dot{\theta }_{q}} \right | \right\},\quad i=1,2,\ldots ,n\end{equation}

Therefore, the problem of solving the robot’s maximum motion velocity $k_{R{max }}{\, \dot{\kern-2pt x}}_{{new}}$ in the specified direction can be transformed into a LP problem for maximizing $k_{{R_{{max }}}}$ :

(35) \begin{align}\begin{array}{c} \max \,F\!\left(t_{i},\alpha ,\beta \right)\\[4pt] \mathrm{s}.\mathrm{t}.\quad \begin{cases} y_{p}\left(t_{i},\alpha ,\beta \right)=0,\\[4pt] -1\leq \dot{\theta }_{q}\left(t_{i},\alpha ,\beta \right)\leq 1 \end{cases} \\[15pt] p=1,2,\ldots ,m-1\\[4pt] q=1,2,\ldots ,n\\[4pt] i=1,2,\ldots ,n-r\!\left(J\right) \end{array} \end{align}

Therefore, given the robot end-effector’s motion parameters ( ${\, \dot{\kern-2pt x}}_{\textit{unitT}},\,{\, \dot{\kern-2pt x}}_{\textit{unitR}},\,d$ ) and joint velocity constraints, the robot’s maximum motion velocity in the specified direction at the target joint configuration can be approximately determined. Notably, we adopt the classical interior-point method for LP, which is particularly effective for small- to medium-scale LP problems, offering minimal computational overhead and strong robustness.

(36) \begin{equation}k_{R\max }=\min _{i}\!\left\{\left | \frac{\dot{\theta }_{q\max }}{\dot{\theta }_{q}}\right | \right\},\quad i=1,2,\ldots ,n\end{equation}

Since the LP method is used for solving, this approach takes more time to compute compared to the DTF method. The advantage of this method lies in the fact that for serial robots with different degrees of freedom, the entire solving process remains unchanged. The only variations throughout the process are the degrees of freedom m of the workspace and the number of joints n of the robot. Therefore, this method is applicable to both nonredundant and redundant robots. Moreover, as shown in Eq. (36), since this method considers the velocity constraints of each joint, it can be applied to scenarios where the joint velocity limits of the serial robot are different.

4.1. Experiment and analysis

In this section, experiments were carried out to verify the accuracy of the proposed method for solving the oriented maximum velocity of serial robots, by comparison with the DTF method. A set of joint configurations is randomly selected first, and multiple directions in the robot’s workspace are then used to measure the maximum translational and rotational velocities. Two robotic platforms, i.e., UR5e (6-DoF) and Sawyer (7-DoF), are utilized in the experiments to evaluate the performance in both nonredundant and redundant serial robots, applied to scenarios where the joint velocity limits of the serial robot are different.

4.2. Experimental design

In the experiment series, the initial joint configurations for the nonredundant robot (i.e., UR5e) experimental joint configuration is selected as [−163.46°, −88.43°, −103.69°, −54.15°, 84.15°, −25.78°], while for the redundant robot (i.e., Sawyer) is [−193.46°, −88.43°, −193.69°, −54.20°, 84.15°, −95.78°, 50°]. For the directions, we used the spherical coordinate system of the end-effector. Namely, with a step size of 10°, the polar angle is taken from 0° to 180°, and the azimuth angle is taken from 0° to 350°, resulting in a total of 648 directions. Since 34 of these directions overlap, the remaining 614 unique directions are selected as the experimental directions after removing the overlapping ones, as illustrated with red arrow in Figure 2.

For the translational velocity, the robot end-effector moves in the direction indicated by the arrow. For angular velocity, its direction is determined by the axis of rotation, and the positive sense follows the right-hand rule: when the right thumb points along this axis vector, the curl of the remaining fingers indicates the positive direction of rotation.

Figure 2.

Experimental directions.

To ensure that the robot reaches its maximum speed upon arriving at the target position, we prescribe an acceleration phase so that it attains the desired peak velocity.

In the experiments, we defined three positions:

1. 1. Target position: the position at which the experiment records the maximum speed to be reported as the result.

2. 2. Start position: the position obtained by moving from the target position for 2 s in the direction opposite to the prescribed direction.

3. 3. End position: the position obtained by moving from the target position for 2 s along the prescribed direction (Figure 3).

Figure 3.

Experimental motion sequence. (a), (b), (c) Translational experiments and (d), (e), (f) rotational experiments.

The experiment procedure involves:

• • Move robot to target position.

• • Move(Revolve) backward at maximum translational(rotational) velocity for 2 s. After reaching the target position, the robot moves for 2 s at its maximum speed in the direction opposite to the experimental direction to arrive at the start position, thereby leaving sufficient room to accelerate to maximum speed upon returning to the target.

• • Move(Revolve) forward at maximum translational(rotational) velocity. Starting from the departure position, the robot moves along the experimental direction at maximum speed, and the speed at the instant it passes the target position is recorded.

• • The motion stops when the robot continues to move and reaches the end position

  (38) \begin{align}e= \bigg| \frac{{\, \dot{\kern-2pt x}}_{e}-{\, \dot{\kern-2pt x}}_{t}}{{\, \dot{\kern-2pt x}}_{e}} \bigg|\end{align}

Where ${\, \dot{\kern-2pt x}}_{e}$ is the experimental velocity and ${\, \dot{\kern-2pt x}}_{t}$ is the theoretical velocity.

4.3. Robotic experimental platform

In this section, the real-world search experiment was verified in the laboratory environment. As shown in Figure 4, the UR5e robot is a six-degree-of-freedom serial robot. The host computer, equipped with an i7-10700 CPU and 16 GB RAM, communicates with the UR5e robot via RTDE over Ethernet. It controls robot motion and receives real-time joint and TCP data at 500 Hz. Joint velocity limits are set via the teach pendant, and all control and data acquisition programs are developed in Python.

Figure 4.

UR 5e robot.

Figure 5.

Sawyer Robot.

As shown in Figure 5, the Sawyer robot is a seven-degree-of-freedom serial robot. The host computer runs Ubuntu with ROS and MoveIt! for controlling the Sawyer robot via Ethernet. It uses the MoveGroupCommander interface for motion planning and executes commands through ROS nodes. Joint limits are set in joint_limits.yaml, and real-time joint and TCP data are collected at 500 Hz. All programs are implemented in Python.

4.4. Experimental results analysis

Based on the aforementioned experimental design, tests were conducted on both robots, and results are analyzed with focus on two aspects: the accuracy of the velocity and the computation time.

In the nonredundant robot experiment, Figure 6(a) and (b) illustrates the distribution of calculated maximum velocity values using the two methods for the 614 selected directions, along with the experimental values. The results show that the proposed method and the DTF method produce identical results (Table I), which is because, in the nonredundant case, the Jacobian matrix of the selected configuration in the experiment is invertible, and the solution corresponding to Eq. (22) is unique. Therefore, the generalized inverse solution used in the DTF method and the fundamental solution set used in this paper yield the same results. The maximum error is 2.50%, with a mean error of 0.44% and a standard deviation of 0.32%.

Table I.

Comparison between DTF and our proposed methods in a nonredundant case (UR5e)

Table I. Long description

The table compares computation error and computation time between DTF and a proposed method in a nonredundant case. It has four rows and four columns. The columns are labeled ‘Computation error’, ‘DTF’, ‘Ours’, ‘Computation time (ms)’, ‘DTF’, and ‘Ours’. The rows are labeled ‘Maximum value’, ‘Minimum value’, ‘Mean value’, and ‘Standard deviation’. Row 1: Maximum value, 2.5 percent, 2.5 percent, 0.15 milliseconds, 4.37 milliseconds. Row 2: Minimum value, 0.01 percent, 0.01 percent, 0.02 milliseconds, 1.06 milliseconds. Row 3: Mean value, 0.44 percent, 0.44 percent, 0.02 milliseconds, 1.38 milliseconds. Row 4: Standard deviation, 0.32 percent, 0.32 percent, 0.01 milliseconds, 0.35 milliseconds.

Figure 6.

Results comparison between two calculated (both DTF and ours) and experimental oriented maximum velocity by a nonredudant robot (UR5e): (a) rotational movement and (b) translational movement.

In terms of computation time, as shown in Figure 7, the overall computation time of the proposed method is higher than that of the DTF method, with average computation times of 1.38 ms and 0.02 ms, respectively (Table I). This is consistent with the theoretical analysis. Unlike nonredundant robots, the proposed algorithm shows significant differences in the accuracy of maximum velocity calculations for redundant robots.

Figure 7.

Computation time comparison between DTF and ours method when performing oriented maximum velocity by a nonredudant robot (UR5e): (a) rotational movement and (b) tansliational movement.

Figure 7. Long description

Two scatter plots compare computation times for rotational and translational movements using two different methods. Panel A: The scatter plot shows computation times for rotational movement. The x-axis is labeled ‘Experiment Direction Index’ and ranges from 0 to 700. The y-axis is labeled ‘log10 Computation Time (ms)’ and ranges from -2 to 1.5. Two data series are plotted: one in blue representing the DTF method and another in green representing the proposed algorithm. The blue data points are generally lower on the y-axis compared to the green data points, indicating lower computation times for the DTF method. Panel B: The scatter plot shows computation times for translational movement. The x-axis is labeled ‘Experiment Direction Index’ and ranges from 0 to 700. The y-axis is labeled ‘log10 Computation Time (ms)’ and ranges from -2 to 1.5. Similar to Panel A, two data series are plotted: one in blue representing the DTF method and another in green representing the proposed algorithm. The blue data points are generally lower on the y-axis compared to the green data points, indicating lower computation times for the DTF method.

Figure 8(a) and (b) shows the distribution of calculated maximum velocity values using the two methods for the 614 selected directions, along with the experimental values. The proposed method has a maximum error of 2.20%, a mean error of 0.67%, and a standard deviation of 0.38%. In contrast, the DTF method shows a maximum error of 24.3%, a mean error of 12.56%, and a standard deviation of 7.04%. Overall, our proposed method performs better than the DTF method. This is because, in the redundant case, the Jacobian matrix is noninvertible and the workspace velocity corresponding to the sum of $\dot{\theta }_{T}$ and $\dot{\theta }_{R}$ , solved using the generalized inverse matrix, does not necessarily represent the maximum workspace velocity among the infinite set of joint velocity solutions, leading to inaccuracies in the solution.

Figure 8.

Results comparison between two calculated (both DTF and ours) and experimental oriented maximum velocity by a redudant robot (Saywyer): (a) rotational movement and (b) tansliational movement.

In terms of computation time, as shown in Figure 9, the proposed algorithm’s experimental results show that the nonredundant robot’s computation time is overall higher than that of the DTF method, which is consistent with the results for the redundant robot. The average computation times are 1.74 ms and 0.04 ms, respectively. Compared to the nonredundant robot, the computation time for both algorithms increases overall when applied to the redundant robot.

Table II.

Comparison between DTF and our proposed methods in a redundant case (Sawyer).

Table II. Long description

The table compares computation error and computation time between DTF and a proposed method for a redundant case. It has two main sections: Computation error and Computation time, each with two columns: DTF and Ours. The table has four rows under each section, labeled Maximum value, Minimum value, Mean value, and Standard deviation. Row 1: Maximum value, Computation error: DTF, 24.3 percent; Ours, 2.2 percent. Computation time: DTF, 5.07 milliseconds; Ours, 24.2 milliseconds. Row 2: Minimum value, Computation error: DTF, 0.22 percent; Ours, 0.01 percent. Computation time: DTF, 0.02 milliseconds; Ours, 1.14 milliseconds. Row 3: Mean value, Computation error: DTF, 12.56 percent; Ours, 0.67 percent. Computation time: DTF, 0.04 milliseconds; Ours, 1.74 milliseconds. Row 4: Standard deviation, Computation error: DTF, 7.04 percent; Ours, 0.38 percent. Computation time: DTF, 0.20 milliseconds; Ours, 1.36 milliseconds.

Figure 9.

Computation time comparison between DTF and ours method when performing oriented maximum velocity by a redudant robot (Sawyer): (a) rotational movement and (b) tanslational movement.

Table I summarizes the performance of the two methods applied to the nonredundant robot. The proposed method transforms the problem of solving the oriented maximum velocity into an equivalent standard LP problem, and the computed result can converge to the optimal solution within the feasible region. Therefore, the calculated result is theoretically the true value, with minimal error compared to the experimental value, and the error mainly originates from experimental inaccuracies. The DTF method’s results closely match those of the proposed algorithm in terms of computation error. This is because the Jacobian matrix is full rank in nonredundant robots at nonsingular positions, and the solution is unique. Theoretical analysis aligns with the experimental results.

Table II shows the performance of the two methods applied to the redundant robot. The proposed method exhibits minor calculation errors compared to the experimental values, and the computational performance is roughly the same as that of the nonredundant robot. However, the DTF method shows a more significant deviation from the experimental results than the proposed algorithm. At nonsingular configurations, regardless of redundancy, the augmented Jacobian (under the respective constraint) is generally noninvertible, but a solution can be obtained using the Moore–Penrose pseudoinverse. The cause of the error in the obtained solution has already been described following Eq. (12) in Section 2.3. The theoretical analysis is consistent with the experimental results.

Tables I and II also show the computation times of the two methods applied to different robots. Since the proposed algorithm uses a LP approach and considers all solutions within the fundamental solution set, the computation time is longer than the DTF method.