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Adaptive Control of Robot Manipulators With Uncertain Kinematics and Dynamics
Hanlei Wang
TL;DR
The paper addresses adaptive tracking for robot manipulators with uncertain kinematics and dynamics. It proposes two adaptive controllers with a separation property, and reports that suitable modifications to the first can improve performance without conservative gain choice.
Problem
The paper studies adaptive task-space tracking for robot manipulators subject to both kinematic and dynamic uncertainties.
Method
The paper proposes two adaptive controllers using a separation approach that separates the kinematic and dynamic loops and defines a joint reference velocity.
Results
The proposed controllers have the separation property, while suitable modifications to the first controller can improve performance without conservative gain choice.
Takeaways & Limitations
Adaptive inverse Jacobian feedback is presented as preferable to commonly used adaptive transpose Jacobian feedback for potentially good task-space tracking performance.
Abstract
from arXiv · showhide
In this paper, we investigate the adaptive control problem for robot manipulators with both the uncertain kinematics and dynamics. We propose two adaptive control schemes to realize the objective of task-space trajectory tracking irrespective of the uncertain kinematics and dynamics. The proposed controllers have the desirable separation property, and we also show that the first adaptive controller with appropriate modifications can yield improved performance, without the expense of conservative gain choice. The performance of the proposed controllers is shown by numerical simulations.
I. INTRODUCTION
The paper addresses adaptive task-space control when both robot kinematics and dynamics are uncertain. It proposes two controllers with separated kinematic and dynamic loops, including a modified first controller intended to improve performance without conservative gain selection.
- Motivation: Transpose Jacobian feedback is reported as undesirable for manipulators moving over a large range.Inverse Jacobian feedback has more involved stability analysis than transpose Jacobian feedback.
- Motivation: Existing adaptive Jacobian methods address uncertain robot control, but performance under simultaneous kinematic and dynamic uncertainty remains unclear.Reported concerns include tracking accuracy and transient response.
- Contributions: The paper proposes two adaptive controllers for uncertain kinematics and dynamics using a separation approach.The separation is realized through a kinematic parameter adaptation law or a new joint reference velocity definition together with the control law.
- Contributions: The proposed separation property requires the joint velocity tracking error to be square-integrable and bounded.The kinematic and dynamic loops remain coupled in many existing results, whereas the proposed approach avoids the approximate transpose Jacobian matrix.
- Contributions: The modified first controller improves closed-loop performance without conservative gain selection and tends to outperform approximate transpose Jacobian feedback under constant gains.The approach extends an earlier scheme to handle both kinematic and dynamic uncertainties.
- Industrial application: The separation property supports industrial joint-velocity control because the joint reference velocity can serve as the servoing command.The adaptive transpose Jacobian approach does not fit this circumstance because of coupling in the torque input.
II. KINEMATICS AND DYNAMICS
The manipulator model combines nonlinear task-space kinematics with standard robot dynamics. Both models are represented as linear in constant parameter vectors, enabling kinematic and dynamic adaptation.
- Kinematics: The end-effector position is related to joint position through a nonlinear mapping, and its time derivative relates task-space velocity to joint-space velocity through the Jacobian.The Jacobian depends on the joint configuration.
- Kinematics: Unknown kinematic parameters prevent direct computation of task-space position and velocity from the direct kinematics.Task-space sensors such as a camera are assumed to provide position and velocity information instead.
- Kinematics: The kinematics depends linearly on a constant parameter vector through a kinematic regressor matrix.The regressor is written as Y_k(q, ξ).
- Dynamics: Manipulator dynamics use an inertia matrix, Coriolis and centrifugal matrix, gravity torque, and joint control torque.The inertia matrix is symmetric and uniformly positive definite, while Ṁ(q) − 2C(q, q̇) is skew-symmetric.
- Dynamics: The dynamics depend linearly on a constant dynamic parameter vector through a dynamic regressor representation.The representation is M(q)ζ̇ + C(q,q̇)ζ + g(q) = Y_d(q,q̇,ζ,ζ̇)a_d.
III. ADAPTIVE CONTROL
The adaptive-control objective is asymptotic task-space trajectory tracking for a robot manipulator. The desired trajectory and its first two derivatives are assumed bounded.
- Control objective: The controller-design objective is to drive the robot end-effector to asymptotically track a desired task-space trajectory.The target condition is x − x_d → 0 as t → ∞.
- Assumptions: The desired task-space trajectory x_d, its velocity ẋ_d, and its acceleration ẍ_d are assumed bounded.This boundedness condition accompanies the asymptotic tracking objective.
A. Adaptive Controller I
Adaptive Controller I combines an estimated-Jacobian joint reference velocity with dynamic and kinematic parameter adaptation. Under a nonsingular estimated Jacobian and revolute joints, task-space position and velocity errors converge to zero.
- The controller defines joint reference velocity from the estimated Jacobian and task-space tracking quantities, then applies adaptive dynamic and kinematic updates.
- Its feedback can be interpreted as inverse-Jacobian feedback for both task-space tracking error and kinematic parameter estimation error.
- The kinematic adaptation regressor depends on joint reference velocity rather than measured joint velocity, while the dynamic adaptation law follows the established formulation.
- The closed-loop dynamics produce a dissipative term −s^T Ks, yielding bounded adaptive estimates and square-integrable, bounded joint tracking error.
- Assuming a nonsingular estimated Jacobian and revolute joints, the controller ensures Δx → 0 and Δẋ → 0 as t → ∞.
B. Adaptive Controller II
Adaptive Controller II retains separation while changing the reference velocity and kinematic adaptation law. It guarantees task-space tracking convergence under a nonsingular estimated Jacobian and also accommodates prismatic joints.
- The second controller uses a different reference velocity and adaptation law while preserving the separation property between kinematic and dynamic loops.
- Theorem 2 states that the controller ensures Δx → 0 and Δẋ → 0 as t → ∞ when the estimated Jacobian is nonsingular.
- Its strong task-space error feedback produces an equivalent velocity-level gain α ˆJ(q) ˆJ^T(q).
- Unlike the first adaptive controller, the second controller is applicable to robots with prismatic joints.
- The separation analysis relies on bounded joint-velocity tracking error, with s ∈ L2 ∩ L∞ and J(q)s ∈ L2 ∩ L∞.
IV. PERFORMANCE OF THE SYSTEM
The paper modifies Adaptive Controller I to improve performance under uncertain kinematics and dynamics. The modification yields certainty-equivalent exponential tracking and avoids conservative gain selection associated with uncertain inertia.
- The modified controller uses estimated-inertia feedback and a corresponding dynamic adaptation-law modification, while retaining the original kinematic adaptation law.
- The gains α and λc quantify response speed and robustness, allowing the desired certainty-equivalent convergence rate γ* by setting λc = α = γ*.
- Theorem 3 guarantees exponential convergence of task-space position tracking error with rate min{λc, α} in the sense of certainty equivalence.
- Under constant-gain feedback, uncertain M(q) generally makes conservative gain selection inevitable for meeting the desired convergence rate.
- Inverse-Jacobian feedback introduces Jacobian-dependent varying gains that compensate for varying task-space inertia more directly than approximate transpose-Jacobian feedback.
V. SIMULATION RESULTS
Numerical simulations compare the proposed controllers with approximate transpose-Jacobian feedback on a 2-DOF planar manipulator. The first controller achieves higher reported tracking accuracy, while the modified version converges faster and more uniformly.
- The simulations use a standard 2-DOF planar manipulator, a 5 ms sampling period, and the stated sinusoidal end-effector trajectory.
- 0.0015 m versus 0.006 m after t = 6 s: the first controller has better tracking accuracy than the controller in,.
- The first controller also provides more adequate utilization of joint torques than the controller in [5], [6].
- After t = 6 s, the second adaptive controller has tracking accuracy comparable to that under the controller in,.
- The estimated-inertia-based modification produces more uniform tracking-error responses and faster convergence than constant-gain feedback in the first controller.
VI. CONCLUSION AND DISCUSSION
The paper addresses adaptive task-space tracking with both kinematic and dynamic uncertainties using two controllers with a separation property. The first controller can achieve improved performance through appropriate modifications, while adaptive inverse-Jacobian feedback is suggested as preferable for task-space tracking.
- The paper considers adaptive tracking for robot manipulators with both uncertain kinematics and dynamics.
- Two proposed adaptive controllers have a separation property between the kinematic and dynamic loops.
- The first adaptive controller can ensure performance under certainty equivalence after modifying its control and dynamic-parameter adaptation laws.
- Adaptive inverse Jacobian feedback is suggested as preferable to adaptive transpose Jacobian feedback for potentially good task-space tracking performance.
- The separation of kinematic and dynamic loops makes a reduced control scheme suitable for industrial robotic applications.