Impedance Control for Manipulators Handling Heavy Payloads

Impedance control, first proposed by Hogan [1], is used to shape the force-response of a manipulator interacting with its environment. In this approach, the contact force, measured by a wrist force/torque sensor, is utilized by the controller to establish a relationship between force and velocity based on the impedance model. The stability of conventional impedance controllers has been extensively analyzed [2, 3, 4, 5]. Lasky et al. [6] introduced an inner/outer loop control scheme to compensate for uncertainties in robot dynamics using a position control algorithm in the inner loop. Adaptive impedance control schemes for robots with uncertain dynamic model parameters were later presented in [7, 8]. The stability of impedance control when dealing with unknown environments has been addressed in works such as [9, 10, 11, 12]. All of these approaches assume that direct measurements of external or contact forces/moments are available.

When a manipulator carries a heavy payload, conventional impedance control can fail to achieve the desired impedance. This failure occurs because the wrist force sensor measures not only external forces but also the inertial and gravitational forces of the payload [13]. Additionally, a heavy payload can significantly alter the manipulator’s dynamics. This is particularly relevant for space manipulators [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], which are designed to handle heavy payloads by leveraging the weightlessness of space. For instance, the space manipulator aboard the International Space Station (ISS) can handle the Space Shuttle orbiter, which weighs approximately 100 metric tons, and employs impedance control for soft berthing with the ISS [27].

Approaches to address non-contact force components measured by a manipulator’s wrist F/M sensor typically focus on compensating for inertial force components using various estimation methods, rather than modifying the impedance control law itself [28, 29]. Fujita and Inoue [30] explored inertial force estimation for cases with pre-planned trajectories. Uchiyama et al. [31] developed a method to estimate external forces and moments using an extended Kalman filter (EKF), though convergence of the EKF depends on the persistent excitation of input signals [32, 33, 34]. Further methods, such as fusing force and accelerometer data via EKF, were proposed in [35] and extended in [34, 36] to estimate both inertial forces and the full set of inertial parameters of the load. However, these techniques require a 6-axis accelerometer. In contrast, our approach modifies the impedance control law itself, eliminating the need for dynamic estimators or accelerometers to account for inertial forces. This method avoids the challenges of input signal excitation and ensures system stability without requiring an additional sensor. Impedance control using inner/outer loops has been employed to emulate manipulators performing contact tasks [37] and spacecraft dynamics [38, 39]. Frequency-domain analysis of impedance control for elastic-jointed industrial manipulators using an inner/outer loop configuration is discussed in [40]. Impedance control schemes for cooperative manipulators have also been explored, with efforts aimed at controlling object/environment interaction forces [41] and managing internal forces in dual-arm cooperative manipulators [42].

The goal of this paper is to present an impedance control scheme for manipulators carrying a heavy payload, without the need for compensating inertial force/moment measurements [3]. We also investigate the limitations of such a scheme. The controller takes force/moment measurements from a sensor installed between the manipulator wrist and payload to establish the dynamic relationship between force error and position error based on a standard mass-damping-spring model. The impedance controller is further enhanced with an inner/outer loop feedback approach, allowing the specification of a general target impedance model. Using robust control theory [43, 44, 45, 46], we demonstrate that the control algorithm can overcome modeling uncertainties. Previous analyses of conventional impedance control show that contact stability imposes a lower bound on the ratio of the desired inertia matrix to the manipulator inertia matrix [47, 48, 49]. However, we show that additional conditions on the payload, manipulator, and desired inertia matrices are required for the stability of impedance control when carrying a heavy payload [3].

This paper is organized as follows: Section 2 introduces the impedance control law for manipulators with heavy payloads. Section 3 further develops the impedance controller using an inner/outer loop control scheme, and Section 3.2 presents a robust version of the controller to handle modeling uncertainties. Finally, Section 4 provides experimental results.

Fig. 1 shows a manipulator carrying a payload with non-negligible mass. We assume that no kinematic singularities are encountered meaning that the robot can always operate in the 6-dimensional task space. A 6-axis force/moment sensor installed at the manipulator wrist to measure the external force applied to the payload. However due to the payload mass, the force sensor signal $\bm{f}^{\prime}_{s}$ contains the component of the generalized external force ${\bm{f}}^{\prime}_{\rm ext}$ superimposed by the gravitational and inertial forces of the payload. Let us assume that $\bm{q}$ is the vector of the manipulator joint angles, vector $\bm{x}$ is the minimal representation of the position and orientation of the payload, $\bm{v}$ and $\bm{\omega}$ are the linear and angular velocities of the payload expressed in the fixed-body frame $\{{\cal C}\}$, and $\bm{\nu}={\rm col}\left(\bm{v},\bm{\omega}\right)$ is the vector of generalized velocity. Then, the following mapping through the manipulator Jacobian matrix ${\bm{J}}^{\prime}(\bm{q})$ is in order

where $\bm{L}(\bm{q})={\rm diag}\left(\bm{1}_{3},\bm{L}_{o}(\bm{q})\right)$ with $\bm{1}_{3}$ being the $3\times 3$ identity matrix, and the transformation matrix $\bm{L}_{o}$ depends on a particular set of parameters used to represent the orientation [50]. The above kinematic mapping can also be written as $\dot{\bm{x}}=\bm{J}(\bm{q})\dot{\bm{q}}$, where $\bm{J}(\bm{q})=\bm{L}^{-1}(\bm{q}){\bm{J}}^{\prime}(\bm{q})$ is called the analytical Jacobian [50]; here we have assumed that no representation singularities of the rotation occurs. According to the virtual work principal, two sets of generalized forces $\bm{f}^{\prime}$ and $\bm{f}$ performing work on $\bm{\nu}$ and $\dot{\bm{x}}$ satisfy $\bm{\nu}^{T}\bm{f}^{\prime}=\dot{\bm{x}}^{T}\bm{f}$, and hence they are related by $\bm{f}_{s}=\bm{L}^{T}(\bm{q})\bm{f}^{\prime}_{s}$. Similarly, $\bm{f}_{\rm ext}=\bm{L}^{T}(\bm{q})\bm{f}^{\prime}_{\bm{e}xt}$.

Now, assume that the manipulator is cut right at its junction to the payload. Note that the interaction force/moment between the two separated systems of the manipulator and the payload are detected by the force/moment sensor. Then, the equation of motion of the payload in the task space can be written by

where

Here, $m_{p}$ and $\bm{I}_{p}$ are the payload mass and inertia tensor, rotation matrix $\bm{R}(\bm{q})$ represents the end-effector attitude, unit vector $\bm{k}$ is aligned with the gravity vector¹¹1If the z-axis of the coordinate frame $\{W\}$ is perfectly parallel to the earth’s gravity vector, then $\bm{k}={\rm col}\left(0,0,-1\right)$. which is expressed in the manipulator’s base frame $\{W\}$, and $g=9.81\;\mbox{m/s}^{2}$. Note that all force terms in the right-hand-side (RHS) of (2) are expressed in the fixed-body coordinate frame $\{C\}$ whose origin coincides with the payload’s center-of-mass. Note that the linear and angular velocities in (3b) can be computed from the manipulator’s joint angles and velocities using the kinematic relation (1), and hence the nonlinear vector $\bm{h}_{p}$, which contains the Coriolis, centrifugal and gravitational terms associated with the payload, can be expressed as a function of the joint angles and velocities, i.e., $\bm{h}_{p}=\bm{h}_{p}(\bm{q},\dot{\bm{q}})$.

It is known that the dynamics model of a manipulator in the task space becomes [50]

where

$\bm{M}_{m}(\bm{q})$ is the Cartesian inertia matrix of the manipulator; the nonlinear vector $\bm{h}_{m}(\bm{q},\dot{\bm{q}})$ contains the Coriolis, centrifugal and gravitational terms; and $\bm{\tau}$ is the vector of joint torques (see Appendix Appendix A: Manipulator & its Payload for details). Assuming a kinematic singularity does not occur, there is an one-to-one correspondence with the joint vector $\bm{\tau}$ and auxiliary input $\bm{u}$. Therefore, in the following derivation, we will take $\bm{u}$ as the control input for the sake of simplicity. In the case of a redundant manipulator, the Jacobin inverse is not unique. In solving the inverse kinematics of redundant manipulators, there exist several joint torque optimization techniques in the literature, one of which is the weighted pseudoinverse solution, which can be used to maximize dynamic or static load capacity of the manipulator [51]. Now, eliminating the interaction force $\bm{f}_{s}$ from (2) and (4), we can express the combined dynamics of the manipulator and the payload by

For the sake of notation simplicity, in the following developments we consider all mass matrices and the associated nonlinear vectors be functions of $\bm{q}$ and $\bm{q},\dot{\bm{q}}$, respectively, without writing their arguments unless otherwise is specified.

The desired impedance model that dynamically balances the external contact force $\bm{f}_{\rm ext}$ is typically chosen as the second-order system [50]

where $\bm{M}_{d}$, $\bm{D}_{d}$ and $\bm{K}_{d}$ are the desired inertia, damping and stiffness, respectively. Here, we assume that the desired motion trajectory, $\bm{x}_{d}$, is specified to be twice differentiable with respect to time. Eliminating the acceleration from (2) and (6) and then from (5a) and (6), we get a set of two equations:

where $\bm{e}=\bm{x}-\bm{x}_{d}$ is the position error. Finally, substituting $\bm{f}_{\rm ext}$ from (7a) into (7b) we obtain the overall impedance control law as

By inspection, one can show that the impedance control law (8) coincides with the conventional impedance control law [50] if $\bm{M}_{p}\equiv\bm{0}$.

From a stability point of view, the fundamental difference between impedance control of manipulators without and with a payload is that, unlike the former case, the latter does not always lead to a stable system. To this end, denote

Then, it is apparent from expressions (8a) and (8b) that the control input remains bounded if matrix $\bm{1}-\bm{\Delta}_{d}$ is not singular, i.e.,

Since $\det(\bm{1}-\bm{\Delta}_{d})=\prod_{i}(1-\lambda_{i}(\bm{\Delta}_{d}))$, we can say that (10) is equivalent to

where $\lambda_{i}(\cdot)$ denotes the $i$th eigenvalue of matrix $(\cdot)$. In the case that the desired inertia is a diagonal matrix with all of its diagonal elements equal to $m_{d}$, i.e., $\bm{M}_{d}=m_{d}\bm{1}_{n}$, the above condition is reduced to

This section describes experimental results obtained from implementation of the proposed impedance control scheme using a robotic manipulator described in [3]. The inertial properties of the manipulator links have been identified [59] as listed in Table 1. The manipulator joints are equipped with torque sensors which are used by an inner loop feedback controller to minimize the joint friction.

A dummy box is mounted on the manipulator wrist. The payload mass and inertia are:

The Cartesian inertia matrix of the manipulator calculated at a nominal position within the workspace used for the tests is:

The force/moment interaction between the manipulator and the payload were measured by a six-axis JR3 force/moment sensor. The impedance controller scheme with inner/outer loop described in Section 3 was developed and implemented using Simulink. Matrix manipulation was performed by using the DSP Blockset of Matlab/Simulink [60]. The Real-Time Workshop package [61] generated portable C code from the Simulink model which was executed on a QNX real-time operating system.

The objective of the first case study is to show that the proposed controller is able to match the manipulator force-response according to the desired impedance even though the force sensor signal is directly incorporated into the controller without compensating for the inertial forces. To this end, the target mass and inertia of the impedance controller is set to be three times as much as the actual mass and inertia of the payload, i.e.,

Fig. 2 shows trajectories of the forces and moments due to external force impulses applied by hand. These trajectories are obtained by compensating for the gravitational and inertial force components of the force/moment sensor output through an off-line process; see Appendix Appendix D: External Force Estimate for the derivation of the external force as a function of $\bm{f}_{s}$ and $\bm{u}$. The subsequent trajectories of the linear and angular velocities of the payload are illustrated in Fig. 3(a). The simulated velocity profiles in response to the same force/moment inputs according to the target impedance model are also depicted in Fig. 3(b). A comparison between trajectories of Figs. 3(a) and 3(b) shows that the impedance controller has succeeded to establish the target impedance characteristic even though the wrist force sensor sees inertial forces of the load. In order to quantify the difference between the simulated velocity and the robot velocity, the root mean square error (RMSE) of the linear velocity is given in percentage as:

Similarly, the RMSE of the angular velocity is calculated to be

In the second case study, the stability of the impedance control system involving contact task is investigated. In this experiment, the manipulator squeezes its payload against a flat horizontal surface. The standard mass-damper-spring model is chosen as the target impedance with parameters according to (38) and

Note that the damping matrix is chosen so as to achieve a well-damped behavior with damping ratio $\zeta=2$. Initially, the manipulator holds the box just a few centimeters above the flat surface. In order to established a contact force between the box and the environment the manipulator is commanded to move along the z-axis to a position obstructed by the flat surface, i.e., $\bm{x}_{d}={\rm col}\left(0,0,x_{d_{z}},0,0,0\right)$ and $\dot{\bm{x}}_{d}=\ddot{\bm{x}}_{d}=\bm{0}$. Fig. 4 illustrates the actual position of the payload along the z-axis versus the desired one. At $t<5$s when no contact is encountered, the controller regulates the robot position to the desired one just above the surface. However, at $t<5$s the robot is commanded to a position so that the box is obstructed by the environment that leads to the contact at $t=6$s. Fig. 5 shows trajectories of the contact force and moment, while Fig. 6 shows trajectories of the linear and angular velocities. The contact force along z-axis, which stabilizes around $23$ N, and the steady-state position error, $5$ cm, are closely related to the corresponding $470$ N/m stiffness component of the enforced impedance. It is clear from Fig. 5 that, in addition to the z-axis force, significant external moment along y-axis is initially built up due to the misalignment of the box surface with respect to the environment. However, the impedance controller responds to the moment and causes the box to slip in a way that eventually leads to significant reduction of the built-up moment.

An impedance controller for manipulators carrying a rigid-body payload has been developed that does not rely on any acceleration measurements in order to compensate for the inertial forces of the payload. The stability and convergence of the impedance controller have been analytically investigated. The results showed that the control input remains bounded provided that the desired inertia was selected to be different from that of the payload. The impedance controller was further developed with utilizing an inner/outer loop approach. This allows specifying the desired impedance model in a general form. The stability analysis revealed that such an impedance control approach has its own limitation; that is to maintain the stability of the closed-loop system, the ratio of payload inertia to the manipulator Cartesian inertia must be lower than a ceratin value (which depends on the feedback gains). It should be noted that this condition is in addition to the well-known condition on the contact stability, which imposes a lower bound limit on the magnitude of the ratio of the desired inertia matrix to the actual effective inertia matrix when the impedance control is implemented digitally or the actuators are of finite bandwidth. Furthermore, robust control theory was employed to modify the proposed impedance controller so that it could compensate for robot modeling uncertainty. Subsequently, the conditions for achieving robust impedance control in the face of modeling uncertainty have been derived. The conditions for this, however, were found to be more stringent than those of the conventional robust position control of manipulators. Experimental results have shown that the proposed impedance controller enabled a manipulator carrying a payload with a non-negligible mass to establish the desired impedance characteristic even though no acceleration measurements were used.