2.1. Multiple redundancies and null-space control

In the case of a robot with 7 DOF, we can decide to describe the primary Cartesian impedance task with only 3 (translational) coordinates ( $m=3$ and $n-m=4$ ). The null-space or secondary task will have up to four DOFs to satisfy or impose its corresponding motions.

In terms of the modal analysis of the redundant system [Reference Saldarriaga, Chakraborty and Kao7], this means that only three modes of motion will be specified for the primary task once all the ( $n-m$ ) ZP modes are successively removed from the system [Reference Kao and Saldarriaga9].

As seen in a recent publication [Reference Hermus, Lachner, Verdi and Hogan16], without analysis or justification, for a given configuration $\mathbf{q}_0$ of a Kuka LBR manipulator, the Cartesian and null-space impedance parameters were chosen for both primary and secondary tasks, including stiffness and damping matrices, without considering inertia reshaping. One of the main claims from the authors of this paper [Reference Hermus, Lachner, Verdi and Hogan16] was that impedance superposition, or simply adding the impedance parameters without any null-space projection matrix, makes little or no difference when comparing the experimental results with those using statically and dynamically consistent null-space projection matrices, especially in cases of more than one redundant degree of freedom.

We will present analysis in the following by applying the theoretical methodology [Reference Saldarriaga, Chakraborty and Kao7, Reference Kao and Saldarriaga9] to show that this claim is only true in special cases depending, among other things, on the chosen sets of parameters or robot configurations, and should not be held as generally valid. The methodology determines the modal natural frequencies ( $\omega _n$ ) and damping ratios ( $\zeta$ ) in the modal space of impedance control, which determines the dynamic responses of the system in the physical space through a linear combination of the modal responses dictated by $\omega _n$ and $\zeta$ , as stipulated by the expansion theorem.

By analyzing the results presented in ref. [Reference Hermus, Lachner, Verdi and Hogan16], the corresponding $\omega _n$ and $\zeta$ for the set of primary and secondary stiffness and damping values that the authors used for 3-DOF Cartesian impedance control (with 4 DOR) is compared in Table I with a dynamically consistent null-space projection matrix $\mathbf{W}=\mathbf{M}(\mathbf{q}_0)$ and superposition of impedances with $\mathbf{N}=\mathbf{I}$ , respectively.Footnote ¹ From Table I, we observe that not all modes are well-damped and the modal $\omega _n$ and $\zeta$ are nearly the same between the corresponding poles of both cases, due to the chosen impedance parameters. This explains the observation of almost identical behavior in the experimental results in ref. [Reference Hermus, Lachner, Verdi and Hogan16], although it is not because of the difference in the null-space controls.

Performing a similar comparison, now with 6-DOF Cartesian impedance control (1 DOR), using the same stiffness and damping values that the authors [Reference Hermus, Lachner, Verdi and Hogan16] used, we can find the results in Table II. We observed that the superposed (when $\mathbf{N}=\mathbf{I}$ ) impedance parameters are changing the dynamics of the system (damping ratios) in a significant manner, especially in the dominant modes of $\lambda _{3,4}$ , $\lambda _{5,6}$ and $\lambda _{7,8}$ , which explains the differences in the experimental results presented in ref. [Reference Hermus, Lachner, Verdi and Hogan16].

Table I.

Comparison of modal $\omega _n$ and $\zeta$ of a 7-DOF Kuka robot with 4 degrees of redundancy, and $\mathbf{W}=\mathbf{M}$ or $\mathbf{N}=\mathbf{I}$ .

Table I. Long description

A table comparing modal and values of a 7-DOF Kuka robot with 4 degrees of redundancy, and or . The table has two cases, each with 7 columns and 2 rows. The columns are labeled with lambda subscripts and the rows are labeled with omega sub n and zeta. Case I: W equals M of q0 has values for lambda sub 1,2 to lambda sub 13,14. Case II: N equals I has values for lambda sub 1,2 to lambda sub 13,14. The values in the table show the comparison of modal and values for the two cases. Notable trends include the similarity in modal and values between the corresponding poles of both cases, indicating nearly identical behavior in experimental results.

In this paper, we will demonstrate how the experimental results and the selection of the parameters can lead to responses that are similar when having different choices of weighting matrix of the null-space control using a redundant robot, and explain in more detail the outcomes of responses. In general, we will show that the choice of different parameters and weighting matrix of the null-space control can affect the responses of redundant robotic manipulators in their corresponding primary and secondary tasks [Reference Kao and Saldarriaga9].

This paper is structured as follows. First, we present an abridged theoretical background of the methodology of modal analysis for impedance control in Section 3. After that, various experimental results using a 7-DOF Franka Emika Panda robot are presented in Section 4. This is followed by a switching strategy of null-space controls to reduce transient responses in Section 5.4. Discussion is presented in Section 5 with insights into the results using impedance control.

Table II.

Comparison of modal $\omega _n$ and $\zeta$ of a 7-DOF Kuka robot with one degree of redundancy, and $\mathbf{W}=\mathbf{M}$ or $\mathbf{N}=\mathbf{I}$ .

Table II. Long description

The table presents a comparison of modal values and damping ratios for a 7-DOF Kuka robot with four degrees of redundancy. It includes two cases, each with seven modal values and seven corresponding damping ratios. Case I shows values such as 1.61, 3.12, 4.72, 8.71, 18.2, 19.2, and 21.6 for the modal values, and 0.213, 0.157, 0.229, 0.0822, 0.175, 0.958, and 0.239 for the damping ratios. Case II shows values such as 1.58, 2.91, 4.53, 9.09, 19.3, 19.2, and 23.3 for the modal values, and 0.207, 0.151, 0.224, 0.121, 0.234, 0.959, and 0.289 for the damping ratios. The table highlights the differences in the dynamics of the system, particularly in the dominant modes.

3.1. Redundancy resolution

Let $m \leq 6$ be the dimension of the task space of an $n$ -DOF serial robot manipulator, $\mathbf{q} \in \mathbb{R}^n$ be the state vector of the joint angles, and $\mathbf{x} \in \mathbb{R}^m$ be the state vector of the task space. The relation between the torques of the joints $\boldsymbol{\tau }$ and the forces at the robot end-effector $\mathbf{f}$ is given by $\boldsymbol{\tau }=\mathbf{J(\mathbf{q})}^T \mathbf{f}$ where $\mathbf{J}(\mathbf{q}) \in \mathbb{R}^{m \times n}$ is the configuration-dependent Jacobian matrix which relates the vector of joint angle rates $\dot {\mathbf{q}}$ of the manipulator to the end-effector pose rate $\dot {\mathbf{x}}$ by $\dot {\mathbf{x}} = \mathbf{J} \dot {\mathbf{q}}$ . We will use $\mathbf{J}$ to denote $\mathbf{J(\mathbf{q})}$ hereafter.

For redundant manipulators where $n \gt m$ , we define $(n-m)$ as the degree of redundancy (DOR). The dimension of the null space of $\mathbf{J}$ is $n-m$ . We can write $\mathbf{f} = (\mathbf{J}^T)^+\boldsymbol{\tau }$ , where $({\cdot})^+$ is the Penrose-Moore inverse, defined to render $\mathbf{J}\mathbf{J}^+=\mathbf{I}$ , and can be computed by

(1) \begin{equation} \mathbf{J}^{+}=\mathbf{W}^{-1} \mathbf{J}^T\left (\mathbf{J} \mathbf{W}^{-1} \mathbf{J}^T\right )^{-1} \end{equation}

where $\mathbf{W} \in \mathbb{R}^{n \times n}$ is an arbitrary weighting matrix. The matrices $\mathbf{J}$ , $\mathbf{J}^T$ , and $(\mathbf{J}^T)^+$ have at most a rank of $m$ . The joint torque vectors in the null space of $(\mathbf{J}^T)^+$ do not affect the wrench generated at the robot end-effector.

The equation $\boldsymbol{\tau }=\mathbf{J}^T \mathbf{f}$ for redundant manipulators can include the null-space control, written as

(2) \begin{equation} \boldsymbol{\tau }=\mathbf{J}^T \mathbf{f} + \mathbf{N}\boldsymbol{\tau }^* \end{equation}

where $\boldsymbol{\tau }^* \in \mathbb{R}^n$ is an arbitrary vector and $\mathbf{N} \in \mathbb{R}^{n \times n}$ is the null-space projection matrix of $(\mathbf{J}^T)^+$ , calculated by

(3) \begin{equation} \mathbf{N}=\mathbf{I} - \mathbf{J}^T(\mathbf{J}^T)^+ \end{equation}

where $\mathbf{I}$ is a $n \times n$ identity matrix. Therefore, we can consider $\mathbf{J}^T \mathbf{f}$ in Eq. (2) as a torque vector required to perform the primary task and $\mathbf{N}\boldsymbol{\tau }^*$ as a torque vector required to perform a feasible secondary task $\boldsymbol{\tau }^*$ , but projected in the null space of the primary task. This term can also be modified to handle several simultaneous lower-priority tasks using successive or augmented methods without interfering with the primary task [Reference Dietrich, Ott and Albu-Schaffer14]. Note that since the secondary tasks are projected in the null space of the primary task, they are performed either fully or partially. From here on, we assume only two tasks, the primary and the secondary tasks, with their corresponding hierarchy, but it can include as many tasks as physically plausible.

In this paper, we consider three different cases of secondary control tasks in Eqs. (1) and (3): (i) dynamically consistent $\mathbf{W}=\mathbf{M}$ , (ii) statically consistent $\mathbf{W}=\mathbf{I}$ , and (iii) superposition $\mathbf{N}=\mathbf{I}$ , where $\mathbf{N}$ is independent of $\mathbf{J}$ and $\mathbf{W}=\mathbf{0}$ .

3.2. Joint-space and task-space impedance control

Impedance control imposes mass-spring-damper behavior on the interaction between the robot and the environment, either in task space or joint space. Let’s consider the robot’s dynamic equations with $\boldsymbol{\tau }=\mathbf{M}_q \ddot {\mathbf{q}}+\mathbf{c}(\mathbf{q}, \dot {\mathbf{q}})+\mathbf{g}(\mathbf{q})+\boldsymbol{\tau }_{\mathrm{ext}}$ with $\mathbf{M}_q$ being the mass matrix of the manipulator, $\mathbf{c}(\mathbf{q}, \dot {\mathbf{q}})$ a vector of Coriolis and Centripetal terms, $\mathbf{g}(\mathbf{q})$ a vector of gravitational terms, and $\boldsymbol{\tau }_{\mathrm{ext}}$ the joint torque. Let’s adopt the impedance control in the joint and task spaces as follows,

(4) \begin{eqnarray} \mathbf{M}_{q} \ddot {\mathbf{q}}+\mathbf{C}_1\dot {\mathbf{q}}+\mathbf{K}_1\bar {\mathbf{q}}&=& \boldsymbol{\tau }_{\mathrm{ext}} \end{eqnarray}

(5) \begin{eqnarray} \boldsymbol{\Gamma }_{C} \ddot {\mathbf{x}}+\mathbf{C}_C\dot {\mathbf{x}}+\mathbf{K}_C\bar {\mathbf{x}} &=& \mathbf{f}_{\mathrm{ext}} \end{eqnarray}

where $\mathbf{M}_{q} \in \mathbb{R}^{n \times n}$ and $\mathbf{\Gamma }_{C} \in \mathbb{R}^{m \times m}$ are joint and task-space mass matrices, respectively, $\mathbf{C}_1 \in \mathbb{R}^{n \times n}$ and $\mathbf{K}_1 \in \mathbb{R}^{n \times n}$ are the damping and stiffness matrices in joint space, $\mathbf{C}_C \in \mathbb{R}^{m \times m}$ and $\mathbf{K}_C \in \mathbb{R}^{m \times m}$ are the damping and stiffness matrices in task space, $\boldsymbol{\tau }_{\mathrm{ext}}$ is external torque at the joints, and $\mathbf{f}_{\mathrm{ext}}$ is external forces applied at the end-effector. The deviations $\bar {\mathbf{q}} = (\mathbf{q}-\mathbf{q}_d)$ and $\bar {\mathbf{x}} = (\mathbf{x}-\mathbf{x}_d)$ are defined here, where $\mathbf{q}_d$ and $\mathbf{x}_d$ are the desired state vectors in the joint and task spaces. Note that $\ddot {\bar {\mathbf{q}}}=\ddot {\mathbf{q}}$ and $\dot {\bar {\mathbf{q}}}=\dot {\mathbf{q}}$ . The same applies to the state vector $\mathbf{x}$ .

From the general definition of Jacobian, the stiffness and damping matrices in Eqs. (4) and (5) are made equivalent by the following equations [Reference Chen and Kao24, Reference Saldarriaga, Chakraborty, Kao, Asfour, Yoshida, Park, Christensen and Khatib25]

(6) \begin{align} {\mathbf{K}}_1&=\mathbf{J}^T\mathbf{K}_C\mathbf{J}+ \mathbf{K}_g +\mathbf{J}^T\mathbf{C}_C\dot {\mathbf{J}} \end{align}

(7) \begin{align} {\mathbf{C}}_1&=\mathbf{J}^T\mathbf{C}_C\mathbf{J} \end{align}

where $\mathbf{K}_g= [ (\tfrac {\partial \mathbf{J}^T}{\partial q_1}\mathbf{f} \,) (\tfrac {\partial \mathbf{J}^T}{\partial q_2}\mathbf{f} \,)\cdots ( {\partial \mathbf{J}^T}{\partial q_n}\mathbf{f} \,) ]$ is the matrix reflecting the changes in geometry under the presence of external forces when transforming between Cartesian and joint spaces [Reference Chen and Kao24], and ${\mathbf{K}}_1$ and ${\mathbf{C}}_1$ are stiffness and damping matrices of the primary task mapped from the Cartesian space into the joint space.

The following torques are calculated to apply to the 7-DOF robot controller in the joint and task spaces, respectively [Reference Ficuciello, Villani and Siciliano26–Reference Stanisic and Fernandez30]Footnote ²

(8) \begin{equation} \boldsymbol{\tau }_q=\mathbf{K}_2 \left (\mathbf{q}-\mathbf{q}_d \right ) + \mathbf{C}_2 \dot {\mathbf{q}}+\mathbf{g}(\mathbf{q}) \end{equation}

(9) \begin{equation} \boldsymbol{\tau }_C=\mathbf{J}^T(\mathbf{q})\left (\mathbf{K}_C\left (\mathbf{x}-\mathbf{x}_d\right ) + \mathbf{C}_C \dot {\mathbf{x}}\right )+\mathbf{g}(\mathbf{q}) \end{equation}

The gravity compensation $\mathbf{g}(\mathbf{q})$ is provided by the control library of the Franka Emika Panda robot, and $\mathbf{K}_2$ and $\mathbf{C}_2$ are the joint stiffness and damping matrices of the secondary task. Consider the primary and secondary tasks defined in Eq. (2), the total torque is

(10) \begin{equation} \boldsymbol{\tau } = \boldsymbol{\tau }_C + \mathbf{N}(\boldsymbol{\tau }_q - \mathbf{g}(\mathbf{q})) \end{equation}

where the null-space projection matrix $\mathbf{N}$ is given in Eq. (3).

Therefore, the control law in Eq. (10) can be rewritten as

(11) \begin{equation} \boldsymbol{\tau } = {\mathbf{K}}_q\left (\mathbf{q}-\mathbf{q}_d\right ) + {\mathbf{C}}_q \dot {\mathbf{q}}+\mathbf{g}(\mathbf{q}) \end{equation}

where ${\mathbf{K}}_q$ and ${\mathbf{C}}_q$ are $n\times n$ stiffness and damping matrices in the joint space, defined as

(12) \begin{equation} {\mathbf{K}}_q = {\mathbf{K}}_1 + \mathbf{N} \mathbf{K}_2, \end{equation}

(13) \begin{equation} {\mathbf{C}}_q = {\mathbf{C}}_1 + \mathbf{N} \mathbf{C}_2. \end{equation}

By substituting the $\boldsymbol{\tau }$ in Eq. (11) into $\boldsymbol{\tau }_{\mathrm{ext}}$ in the general equation of motion of a robot in the joint space, the impedance control can be analyzed and solved.

3.3. Modulation of dynamic response through stiffness and damping in primary and secondary tasks

To obtain a desired dynamic behavior, we can use a closed-form solution that allows us to modulate the response through stiffness and damping [Reference Kao and Saldarriaga9]. This analytical tool gives us insights into how the parameters of the impedance controller affect the overall response based on the results of the modal natural frequencies and damping ratios.

Before solving systems with redundancy, we need to first remove the ZP motions and obtain the response for the remaining non-ZP (NZP) motions in the modal space [Reference Saldarriaga, Chakraborty and Kao7, Reference Kao and Saldarriaga9, Reference Saldarriaga and Kao10]. Next, we can obtain the dynamic responses of the entire system in the form of Eq. (4) by considering one or more tasks, as expressed by Eqs. (12) and (13), in the modal space using the following state-space equation of the linear system [Reference Kao and Saldarriaga9]

(14) \begin{equation} \dot {\mathbf{z}}(t)=\mathbf{A}\,\mathbf{z}(t)+\mathbf{B}\,\boldsymbol{\tau } \end{equation}

where $\mathbf{z} = [\bar {\mathbf{q}}^T,\dot {\bar {\mathbf{q}}}^T]^T \in \mathbb{R}^{2n}$ and $\mathbf{A} \in \mathbb{R}^{2n \times 2n}$ and $\mathbf{B} \in \mathbb{R}^{2n \times n}$ are given as

(15) \begin{equation} \mathbf{A} = \begin{bmatrix} \mathbf{0} & \mathbf{I} \\ -\mathbf{M}^{-1}{\mathbf{K}_q} & -\mathbf{M}^{-1}{\mathbf{C}_q}\\ \end{bmatrix}\!, \, \mathbf{B}=\left [ \begin{array}{c} \mathbf{0}\\ \mathbf{M}^{-1} \end{array} \right ] \end{equation}

The solution of the linear system in Eq. (14) is

(16) \begin{eqnarray} \mathbf{z}(t) = \mathbf{X}e^{\mathbf{\Lambda }t}\mathbf{Y}^T\mathbf{z}(0) + \int _{0}^{t}\mathbf{X}e^{\mathbf{\Lambda }\xi }\mathbf{Y}^T\mathbf{B}\boldsymbol{\tau }(t-\xi )d\xi \end{eqnarray}

where $\mathbf{X}$ and $\mathbf{Y}$ are matrices containing the right and left eigenvectors of the state-transition matrix $\mathbf{A}$ , respectively, and $\boldsymbol{\Lambda }$ is a diagonal matrix with the corresponding eigenvalues of $\mathbf{A}$ .

By applying this methodology, we can solve the equations analytically for different values of impedance parameters to obtain a trend or insights into how the response is being affected, without trial and error. In addition, the methodology allows us to analyze and modulate the responses by taking into consideration all primary and secondary null-space tasks, with or without the ZP motions.

4.1. 7-DOF robot with one degree of redundancy

We first consider the case of the Cartesian impedance controller in the primary task with 6 DOFs ( $m=6$ ). Since the Panda robot has 7 DOFs ( $n=7$ ), this leaves $(n-m)=1$ degree of redundancy for secondary null-space tasks of joint impedance control.

Experiments were conducted using the following matrices (all in SI units), obtained through the modal analysis, to trace a circular trajectory in the XY plane of a 12-cm radius. The weighting matrix $\mathbf{W}$ in Eq. (1) is the mass matrix $\mathbf{M}$ .

• • $\mathbf{K}_C=$ diag $(4000, 2000, 2000, 100, 100, 100)$

• • $\mathbf{C}_C =$ diag $(346.4, 124.3, 146.1, 1.35, 2.3, 0.26)$

• • $\mathbf{K}_2=$ 30 $\mathbf{I}$ Footnote ³

• • $\mathbf{C}_2=$ diag $(0.1, 0.1, 3.3, 0.1, 4, 0.1, 0.1)$

The corresponding natural frequencies and damping ratios in the modal space are listed in Table III.

Table III.

Modal $\omega _n$ and $\zeta$ of a 7-DOF Panda robot for given Cartesian and secondary impedance parameters.

Table III. Long description

The table presents the natural frequencies and damping ratios in the modal space for a 7-DOF Panda robot with specified Cartesian and secondary impedance parameters. It consists of three rows and seven columns. The first row contains the headers for the natural frequencies labeled as lambda subscript 12, lambda subscript 34, lambda subscript 56, lambda subscript 78, lambda subscript 910, lambda subscript 1112, and lambda subscript 1314. The second row, labeled as omega n, lists the values 2.7, 3.82, 4.1, 46.3, 2.6, 37.2, and 4.0. The third row, labeled as zeta, lists the values 0.594, 0.70, 0.59, 0.73, 1, 1, and 0.736. The table highlights the varying natural frequencies and damping ratios across different modes, with notable high values in the lambda subscript 78 and lambda subscript 1112 columns.

Figure 1.

Experimental results of trajectories for a redundant robot with one DOR.

Figure 1. Long description

The image contains two graphs. The left graph shows circular trajectories of a robot with a radius of 12 centimeters. The trajectories are plotted for a period of 2 seconds. The graph compares modal analysis with arbitrary damping matrices, represented by different colored lines: black for modal, red for 1 over Kc, and blue for the square root of Kc. The intended circular path is shown as a black dashed line. The right graph displays the errors of different primary task damping matrices over time, with the same color scheme. The x-axis represents time in seconds, and the y-axis represents the error in position. The graphs illustrate how different damping matrices affect the robot’s trajectory and accuracy.

The eigenvalue pairs $\lambda _{9,10}$ and $\lambda _{11,12}$ in Table III are overdamped and they are both on the real axis of the complex plane. Thus, there are two natural frequencies. The dominating (lower) $\omega _n$ is listed in the table.

Next, we compare the experimental results of responses of the system using the damping matrix obtained through modal analysis versus arbitrarily chosen ones, such as $\mathbf{C}_C=\frac {1}{100}\mathbf{K}_C$ , and $\sqrt {\mathbf{K}_C}$ . The experimental results of trajectories of the parameters chosen by the modal analysis, as well as those chosen arbitrarily, are plotted for comparison in Figure 1(a). In the figure, the first two cycles are plotted with a period of 2 s. The intended circular path is plotted in black dashed line. A portion of the trajectory in the first quadrant is shown in zoom-in view for comparison. The norm of errors from the prescribed circular trajectory is plotted in Figure 1(b). When using proper damping from the modal analysis, the errors tend to stabilize or have a median value around zero. The mean error values for each response in Figure 1(b) are $0.0002, -0.0049,$ and $-0.0041$ for the modal, $\frac {1}{100}\mathbf{K}_C$ , and $\sqrt {\mathbf{K}_C}$ damping matrices, respectively. Several experiments with different radii were conducted with similar results.

In addition to better trajectory following demonstrated in Figure 1(b), the control efforts of joints are analyzed and compared. The following observations are in order.

• • The torques, or the control efforts, as a function of time, show that the modal analysis method performs better with smaller control efforts to achieve better trajectory following in all joints. A sample control effort of joint 2 is plotted in Figure 2 for illustration. In comparison, the control effort when applying appropriately chosen damping matrix using modal analysis is always smaller.

• • The integration of control efforts over time is summarized in Table IV which indicates that the impedance control system using the modal analysis has considerably less integration of control efforts in joints 1 to 4 and 6, and are comparable in joints 5 and 7. Note that joint 7 is the joint of the end effector.

Figure 2.

Comparison of torques of joint 2 when performing the trajectory following in Figure 1.

Figure 2. Long description

A line graph compares torques of joint 2 when performing the trajectory following, with three data lines labeled Modal, 1 over 100 K c, and square root of K c. The x axis represents time in seconds ranging from 0 to 6 seconds, and the y axis represents torque in Newton-meters ranging from negative 25 to positive 15. The Modal line fluctuates between negative 20 and positive 10 Newton-meters. The 1 over 100 K c line shows more pronounced oscillations, while the square root of K c line follows a similar pattern to the Modal line but with less amplitude. All values are approximated.

4.2. 7-DOF robot with multiple DOR

In this case, we consider only the three translational components of Cartesian impedance in the primary task with $m=3$ . The secondary impedance control in the joint space has $(n-m)=4$ DOR for the 7-DOF Panda robot.

Table IV.

Integration of joint torques over time for trajectory tracking in Figure 1.

Table IV. Long description

A table with seven columns and four rows comparing the integration of joint torques over time for trajectory tracking across different methods. The columns are labeled with tau subscript 1 through tau subscript 7, representing different joints. The rows are labeled as Modal, one over one hundred K subscript C, and square root of K subscript C. The table shows the values of joint torques for each method. For tau subscript 1, the values are 10.97 for Modal, 11.52 for one over one hundred K subscript C, and 11.64 for square root of K subscript C. For tau subscript 2, the values are 22.03 for Modal, 30.17 for one over one hundred K subscript C, and 26.39 for square root of K subscript C. For tau subscript 3, the values are 8.435 for Modal, 9.463 for one over one hundred K subscript C, and 9.752 for square root of K subscript C. For tau subscript 4, the values are 9.588 for Modal, 10.61 for one over one hundred K subscript C, and 10.64 for square root of K subscript C. For tau subscript 5, the values are 4.652 for Modal, 4.642 for one over one hundred K subscript C, and 4.859 for square root of K subscript C. For tau subscript 6, the values are 3.229 for Modal, 3.676 for one over one hundred K subscript C, and 4.548 for square root of K subscript C. For tau subscript 7, the values are 2.419 for Modal, 2.387 for one over one hundred K subscript C, and 2.952 for square root of K subscript C.

Experiments were conducted using the following matrices, obtained through modal analysis, to trace a circle of 12-cm radius using a weighting matrix $\mathbf{W}=\mathbf{M}$ . For the sake of comparison, we also include a case in which we chop off the rotational components of $\mathbf{C}_c$ directly from the previous case of 1 DOR.

• • $\mathbf{K}_C=$ diag $(4000, 2000, 2000)$

• • $\mathbf{C}_C =$ diag $(54.75, 74, 150.8)$

• • $\mathbf{K}_2=$ 30 $\mathbf{I}$

• • $\mathbf{C}_2=$ diag $(1.75, 0.1, 2.2, 0.1, 1.06, 2, 0.22)$

The corresponding modal natural frequencies and damping ratios from the modal analysis are listed in Table V.

Table V.

Modal $\omega _n$ and $\zeta$ of a 7-DOF Panda robot for given Cartesian and secondary impedance parameters.

Table V. Long description

The table presents the modal natural frequencies and damping ratios for a 7-DOF Panda robot under specific Cartesian and secondary impedance parameters. It consists of three columns labeled lambda 12, lambda 34, lambda 56, lambda 78, lambda 910, lambda 1112, and lambda 1314, and two rows labeled omega n and zeta. The values in the table are as follows: Row 1: lambda 12, 3.52; lambda 34, 6.63; lambda 56, 7.0; lambda 78, 3.42; lambda 910, 3.97; lambda 1112, 6.88; lambda 1314, 30.1. Row 2: lambda 12, 0.646; lambda 34, 0.646; lambda 56, 0.808; lambda 78, 0.626; lambda 910, 0.782; lambda 1112, 0.788; lambda 1314, 0.826. The table provides a detailed comparison of the modal properties of the robot under the specified conditions.

The modal analysis must be conducted using four DOR, instead of chopping off the three rotational DOFs from the matrices. Simply removing rotational elements from previously modulated systems does not yield good results.

The experimental results of trajectories and errors are similar to those shown in the previous section and Figure 1. Several experiments with different radii were conducted with similar results also consistently showing greater control efforts when using several arbitrary damping matrices.

4.3. Dynamically and statically consistent projection matrices versus superposition of impedances

In this section, we study experimentally the responses by applying the dynamically and statically consistent projection matrices when $\mathbf{W=M}$ and $\mathbf{W=I}$ , respectively, in Eq. (1) versus the superposition of the primary and secondary tasks when $\mathbf{N=I}$ , for the case of 4 DOR and a radius of $5$ cm for the circular path. The following matrices are used.

• • $\mathbf{K}_C=$ diag $(4000, 2000, 2000)$

• • $\mathbf{C}_C =$ diag $(50, 54, 162)$

• • $\mathbf{K}_2=$ 30 $\mathbf{I}$

• • $\mathbf{C}_2=$ diag $(0.1, 0.3, 3.87, 1, 1.06, 2.2, 0.2)$

The corresponding modal natural frequencies and damping ratios are listed in Table VI. The responses with different weighting matrix $\mathbf{W}$ and a case of superposition of impedances (when $\mathbf{N=I}$ ) are plotted in Figure 3 for comparison.

Table VI.

Modal $\omega _n$ and $\zeta$ of a 7-DOF Panda robot for given Cartesian and secondary impedance parameters.

Table VI. Long description

The table presents modal natural frequencies and damping ratios for a 7-DOF Panda robot under various impedance parameters. It consists of 4 rows and 8 columns. The columns are labeled with lambda subscripts ranging from lambda 1,2 to lambda 13,14 and the rows are labeled with omega n and zeta. The values in the table indicate the specific natural frequencies and damping ratios for each modal parameter. Notable trends include varying frequencies and damping ratios across different impedance settings, with some values significantly higher than others.

Figure 3.

Experimental results of trajectories for a redundant robot with four DOR using different projection matrices.

Figure 3. Long description

The image contains two graphs. The left graph shows circular trajectories with a radius of five centimeters, comparing dynamically consistent and statically consistent null-space projection matrices. The trajectories are plotted for the first two cycles of motion, with the transient response in the solid black line representing only part of the first cycle. The right graph displays the errors of different null-space projection matrices over time, with the radius set to five centimeters. The errors are plotted for dynamically consistent, statically consistent, and no null-space projection matrix for the secondary task. The graphs use different line styles and colors to distinguish between the matrices: dashed red for W equals I, solid black for W equals M, and dash-dotted blue for N equals I. The x-axis of the left graph represents the X coordinate, and the y-axis represents the Y coordinate. The x-axis of the right graph represents time in seconds, and the y-axis represents the error. The graphs illustrate the performance and accuracy of different projection matrices in controlling the robot’s movements.

A few observations are in order.

1. 1. The superposition of impedances ( $\mathbf{N}=\mathbf{I}$ ), or simply adding the matrices without considering a null-space projection matrix $\mathbf{N}$ of the secondary tasks, is not sound. As expected from theory, it significantly changes the dynamics of the system according to the parameters used.

2. 2. The dynamically consistent null-space projection matrix, with $\mathbf{W=M}$ , generates a response with noticeable transient as shown in Figure 3(a) and (b) in black solid lines. These transients may not be suitable for most applications. This transient behavior was also observed and presented in ref. [Reference Dietrich, Ott and Albu-Schaffer14].

3. 3. When the weighting matrix is chosen as the statically consistent projection with $\mathbf{W=I}$ , the response does not have an initial transient response, although it is not as accurate when compared with the steady-state response of $\mathbf{W=M}$ .

4. 4. Through the proposed modal methodology, we can reduce the transient through a different primary Cartesian damping matrix.Footnote ⁴ The corresponding modal $\omega _n$ and $\zeta$ are listed in Table VII. As expected, a matrix $\mathbf{C}_c$ with higher damping generates a smaller transient response but a higher steady-state error. Nevertheless, the magnitudes of the transient, although reduced, are still too large to eliminate.

Table VII.

Modal $\omega _n$ and $\zeta$ of a 7-DOF Panda robot for given Cartesian and secondary impedance parameters.

Table VII. Long description

The table presents modal and impedance parameters for a 7-DOF Panda robot, focusing on Cartesian and secondary impedance parameters. It consists of three columns labeled lambda subscript 12, lambda subscript 34, lambda subscript 56, lambda subscript 78, lambda subscript 910, lambda subscript 1112, and lambda subscript 1314, and three rows labeled omega subscript n and zeta. The values in the table indicate specific impedance parameters for different degrees of freedom of the robot. Notable trends include varying impedance values across different degrees of freedom, with some values significantly higher than others, suggesting differences in the robot’s response characteristics.