Configuration Optimization for End-Point Stabilization of Redundant Manipulators With Base Flexibility

There are myriad potential health risks due to elevated radiation at a nuclear disaster site. The use of remotely operated robots for tasks such as measurement and reorganization of damaged objects at the site, instead of human responders, is a preferred option. Nuclear plant inspection robots were extensively studied back in the 1980s [1]. Since then, a number of rescue and first-responder robots have been developed in various forms.

On the other hand, in the short-term, technology is required to prepare for a possible case where an accident has occurred and functional robotic systems must be deployed as soon as possible. To address the hardware reliability issues, the current project will develop interfaces that turn existing construction equipment into a mobile robotic manipulator as shown in Fig. 1. One of the authors’ groups has developed wirelessly operated mechanical interfaces that operate the controls of the conventional excavator.

A typical excavator consists of an arm with a gripper or bucket attached to its end, cabin, and undercarriage. The undercarriage of the tracked construction equipment is the base frame that supports all the components of the vehicle. Despite high maneuverability on a rough terrain, the undercarriage introduces flexibility between the ground and upper part of the equipment. This issue is observed as the residual oscillation of the arm end-point. Motion of the upper mechanism, change in the arm posture or cabin rotation, dynamically induces the undercarriage flexibility, resulting in errors in end-point positioning.

When it comes to end-point stabilization of the excavator’s arm in the absolute coordinate system, the undercarriage oscillation and resultant end-point error must be effectively suppressed. The undercarriage oscillation is passive and induced by the rest of the excavator’s motion. There is no actuation system in the general undercarriage to implement vibration suppression control even though its oscillation is measurable, so it is not possible to suppress the vibrations using an active method [2]. However, if simple end-point feedback stabilization control is applied without explicitly modeling the undercarriage flexibility, the bandwidth of the feedback system is significantly limited due to the un-modeled disturbance oscillation [3,4]. This goes on to exclude passive suppression methods that require characterization of the disturbance oscillation like tuned-mass-damper systems [5] and the use of pre-compensators like adaptive input shaping [6]. Current approaches also rely on the characterization of base oscillation parameters or range of oscillation frequency [7,8], fuzzy controllers to compensate for unknown dynamics [9], or sliding mode control [10].

The objective of the paper is to improve the performance of end-point positioning of the excavator boom (or arm) in the absolute coordinate system for manipulation tasks. End-point feedback control will be done without measuring or modeling of induced base oscillations for practical advantages. To improve the robustness of such simple controller implementation, configuration-based stability analysis of a manipulator with an oscillatory base is performed. Once stability margins of the manipulator workspace have been characterized, it is possible to determine the position of the undercarriage and the posture of the arm such that the task-space feedback control has greater robustness to induced oscillation.

Examining a two-degree-of-freedom (2DOF) planar manipulator on a base that is flexible in three directions, shown in Fig. 2, where $θ1$ is the base rotation, $θ2$ and $θ3$ are manipulator joint angles. In the following analysis, the position of the end-point of the arm is considered, and its orientation is not considered. Generally, excavators have an additional degree-of-freedom between the arm and bucket which can be used to compensate for the orientation.

The dynamics of the planar manipulator can be described by $τ=M(q)q¨+H(q,q˙)+N(q,q0,q˙)+G(q)$⁠. $τ$ is is the joint torque vector for the system, consisting of $[τpT,τaT]T$⁠, which are the passive joint torques and the actuator effort torques for the driven joints; the matrix M(q) is the inertia matrix for the system; and $H(q,q˙)$ is the collection of Coriolis factors and centrifugal terms for the manipulator. $N(q,q0,q˙)$ is the general viscoelastic term and G(q) is the gravity term. The state vector, $q=[qpTqaT]T$⁠, contains state information for all of the joints in the system, containing $qp=[xmobile,ymobile,θ1]T$ and $qa=[θ2θ3]T$⁠. Note that the horizontal position of the undercarriage center x_mobile is a variable, but assumed to be fixed when the system is in motion; therefore, it is not included in the dynamic equation.

The practical control system studied in this paper is developed based on the following assumptions: (1) nominal kinematic parameters of the arm are known; (2) nominal dynamic parameters of the arm and undercarriage are known; (3) desired end-point O_ed is given from the operator; (4) end-point O_ed is measurable; (5) arm joint angles q_a are measurable; and (6) undercarriage deformations q_p are not measurable. Inertial measurement units and differential global positioning systems are attached to the excavator body to add sensing capabilities.

Without external odometry, using only forward kinematics is unable to correctly position the end point. However, due to practical difficulties in integrating oscillation measurements and modeling into a dynamic deformation model of the undercarriage, the excavator arm will be treated as a general multi-link manipulator fixed to the ground. Compensation through task-space control is used to reduce end-point positioning error.

Given that any mechanical manipulators have inherent flexibilities in links, joints, and base, a theoretically correct approach would be to implement a controller that compensates for flexibility-oriented deformations by modeling or measuring of them. This approach, naturally, requires additional sensing components and associated modeling, integration and tuning efforts. In contrast, in many practical implementations, manipulators are considered completely rigid and resonant frequencies are significantly higher than the control bandwidth therefore not excited. This approach allows for modeling and control theories developed for rigid manipulators, which is a clear implementation advantage. One of the simplest class of manipulator end-point feedback controllers can be obtained from its Jacobian matrix, derived only from kinematic information, in the form of its inverse or transpose, where dynamic close-loop stability is guaranteed for rigid systems.

The remote excavator system is one of many exceptions where mechanical flexibilities may impact the stability and performance of end-point feedback control. Oscillations due to its flexible undercarriage may be within the control bandwidth and thus introduce oscillatory or unstable responses. As a practical approach, one may still want to use the rigid-body assumption and examine stability and performance limits when applied to a flexible-base manipulator without increasing the complexity of feedback control. Figure 3 illustrates this concept where the base deformation is strategically neglected. The rigid body Jacobian, J_v, can be found by only considering the motion of the two links in the manipulator chain $Jv=[−l2S12−l3S123−l3S123l2C12+l3C123l3C123]$⁠.

The actuated joint torques may be driven by Jacobian-transpose task-space control, $τa=Jv(qa)Tf(Oed,Oe)$⁠, with structure shown in Fig. 4(b). J_v(q_a) is the rigid-body manipulator Jacobian; O_e is the end-point position; O_ed is the desired end-point position of the manipulator; and f(O_ed, O_e) is a task-space feedback controller. This control is applicable when the torque of the manipulator, $τa$⁠, can be directly specified. In other words, hydraulic systems that are used in many excavators need to be modified to allow direct torque control. The stability analysis presented in this paper does not depend on the choice of a feedback control scheme. By applying robust control theory, we consider all stabilizing controllers in the task-space. One could apply simple proportional-derivative control where $f(Oed,Oe)=Kp(Oed−Oe)+Kv(O˙ed−O˙e)$ as a popular choice. K_p and K_v are proportional and derivative diagonal gain matrices, respectively. Gravity compensation may be added to compensate for the offset due to gravity.

Another method to control the manipulator is to specify joint velocities: $q˙ad=Jv(qa)−1f(Oed,Oe)$⁠, with control structure detailed in Fig. 4(a). Note that the majority of construction equipment map the tilt angle of the control lever to the velocity of one of the hydraulic cylinders [11]. As a result, an excavator equipped with the GHOST system can naturally introduce this Jacobian inverse control. Obtained $q˙ad$ is directly converted into control lever displacements realized by the GHOST system, $τa=[1001]q˙ad$⁠.

By using modal analysis, P(s) can be represented as a linear sum of n rigid modes and m vibration modes. Matrix A has in total 2(n + m) poles where 2n of these poles are zero, corresponding to the rigid modes, and 2m conjugate complex poles correspond to the vibration modes.

In Eqs. (3)–(6), R₀ corresponds to the rigid body modes where R₀ is a positive definite matrix. $M^=[Mkl]∈ℜn×n(m+1≤k,l≤m+n)$ is a partial matrix of M. R_i is called a residue matrix. Note that R_i is positive-semi-definite and rank (R_i) = 1 at most from (6). In addition, $ω^i=|λi|,ζi=−Re(λi)/|λi|$ is obtained where $ζi$ and $ω^i$ represent the damping coefficient and the natural frequency, respectively.

Link parameters of the excavator are found in Table 1 and are derived from the geometry of a 3D model assuming the material is steel. The viscoelastic parameters are chosen given the data collected by one of the co-authors’ group.

Robustness analysis is performed to determine the configurations of a manipulator that are robust to undercarriage oscillation and, therefore, are stable. Calculating the H_∞ coprime factorization of the plant across the workspace of the manipulator it becomes possible to characterize the robustness of the plant $εmax$⁠. This analysis, while it does characterize closed-loop stability, does not require the specification of a controller and is gain independent (Appendix  A).

As shown in Fig. 6, $εmax$ is calculated across the workspace where higher values are indicative of more robust configurations. While the robustness is dependent on both the plant model and the servo gain used in the analysis, there is a clear difference in performance between locations with high and low stability margin. Similarly, in Fig. 7, a surface can be generated for each vibrational mode in the system. It can be seen that features in the modal-based robustness surfaces appear in the coprime factorization surfaces. The relationship between the comprime factor-based robustness metric and the modal stability indices can be seen in the GAP metric between the rigid body excavator system and the total flexible system. Utilizing this relationship becomes possible to only use the modal analysis w_i metric to perform posture optimization [3].

The manipulator control system with a flexible base is “colocated” as flexible components are not between the sensing point (end-point) and active joints, but outside of the rigid system. While the colocation of the sensor and actuator forms an “in-phase” or minimum-phase system for a linear 1-DOF dynamic system, its extension to multi-DOF systems is not straightforward. Indeed, colocated flexible control systems may still have a stability problem due to dynamic coupling between axes [3] (i.e., between the x and y axes in the planar dynamic system considered in this paper).

This dynamic coupling creates a feed-through loop in which flexible motion along one axis excites flexible motion along another axis. The excited motion introduces end-point errors to be compensated by the end-point feedback controller, which will further excite base oscillation. This cycle can lead to stability problems in the manipulator system. The degree of coupling between the axes is represented by a residue matrix of each mode, R_i, i = 1, 2, …, as a function of the arm posture. By choosing special postures where R_i indicates no dynamic coupling between axes, it would be possible to design a control system that stabilizes the end-point and is robust to the flexibility of the undercarriage.

Due to the manipulator being on a mobile base, there is a planar redundancy between the x-axis position in absolute space and the x-axis position of the manipulator end-point in the manipulator’s workspace. It is then possible to reconfigure the position of the manipulator’s mobile base in order to put the manipulator’s goal position in a position with a higher stability margin. The variation of stability margin with end-point position for a given y_ed is shown in Fig. 8. Mathematically, can be represented as $q~d=argmaxqd∈{Θ:ye=yd}V(qd)$⁠, where Θ is a set of arm configurations that realizes O_ed [13]. Here, V(q_d) is defined as $V(qd)=∑i=1mαiwi(qd)$⁠.

To maximize computational ease, the use of w₁ will be used to optimize the base position under Jacobian nullspace control for redundant manipulators. While both robustness metrics used in this work are valid for optimization, due to the nature of $εmax$⁠, the computation of its gradient is nontrivial. Furthermore, while $εmax$ measures the robustness of the given plant to modeling errors and disturbances, the w_i metric serves to provide a relationship for a plant robustness to excite its vibrational modes. From Fig. 8, it can be seen that the curves for $εmax$ and w₁ follow the same general shape, so there is not a clear performance increase between using one or the other. The update law for the redundant base parameters $[xmobile,d,θ2d,θ3d]$ is $Δqd=S(I−JB+JB)kξξ,S=[1O3x2O4x1I2x2]$⁠. $kξ$ is a positive gain matrix used to selectively damp the augmentation of the nullspace for each coordinate, and $ξ$ is the gradient of the objective function, Eq. (4). The Jacobian used to map the redundant parameters to the manipulator end-point, J_B, is defined as $JB∈ℜ2x3=[10Jv]$⁠.

Simulations of the nonlinear manipulator system were conducted in matlab (Mathworks) using an RK4 numerical integrator. The integration step of the matlab simulation was selected to be 0.005 s (i.e., 200 Hz), which is significantly higher than the resonant frequencies of the flexible system and its closed-loop control bandwidth. Changing the end-point position in the workspace, one can improve the stability of the end-point control of the manipulator system as illustrated in Fig. 8. Applying both $εmax$ and w_i stability margins to two discrete points in the manipulator workspace, shown in Table 2, we can see an improvement in convergence and when going from a position of low robustness to high robustness. The results of optimizing the mobile base position with Jacobian inverse control are shown in Fig. 9 using two discrete positions, shown in Table 2 based on their w₁, shown in Fig. 7(d). It can be seen that there is better performance for the optimized position that has a higher robustness.

The same phenomena can be seen when moving the manipulator mobile base to change the position of the end-point in the workspace without changing its position in the inertial coordinate space. This can be seen in Figs. 11 and 12. Moving the manipulator base according to the trajectory generated through nullspace augmentation, shown in Fig. 10, does excite some base oscillation and end-point error. However, by moving the system from a configuration of low robustness to one of high robustness, manipulator stability can be achieved.

The use of coprime factorization to generate a stability margin for the manipulator across its workspace provides a means to characterize manipulator configurations that will provide robustness to the flexibilities inherent in its base. While this material can be useful in offline applications, because the stability margin is calculated across the workspace, it can be used in conjunction with optimal control systems in order to minimize the amount of time the manipulator has to occupy configurations with a low stability margin. The occupation of a configuration with a high stability margin allows for the use of a controller with a higher feedback gain.

This paper presented a method for configuration optimization of a rigid manipulator with a flexible mobile platform. The use of the coprime factorization-based stability margin and modal decomposition-based stability margin allows for end-point feedback control robust to model variation in base oscillations. When employed alongside the redundancy introduced by the mobile base of the excavator, configuration optimization is able to position the end-point of the excavator at the desired target position while still utilizing a robust arm configuration without explicitly including the undercarriage flexibility in the controller. As long as the configuration for the manipulator is chosen to have a high stability margin against induced undercarriage oscillation, the feedback system is stable and does not excite unstable vibration modes. Future work includes the extension of this method to experimental validation and to posture-based optimization for contact stability.

This material is based upon work supported by the Air Force Office of Scientific Research (Award No. FA2386-17-1-4655). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force.

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