Optimal control of differentially flat underactuated planar robots in the perspective of oscillation mitigation

Stefano Lovato; Michele Tonan; Matteo Bottin; Matteo Massaro; Alberto Doria; Giulio Rosati

cs.ROMar 16, 2026

Optimal control of differentially flat underactuated planar robots in the perspective of oscillation mitigation

Authors: Stefano Lovato, Michele Tonan, Matteo Bottin, Matteo Massaro, Alberto Doria, Giulio Rosati

Organizations: Department of Industrial Engineering, University of Padova, Via Venezia 1, Padova, Italy

Abstract

Underactuated robots are characterized by a larger number of degrees of freedom than actuators and if they are designed with a specific mass distribution, they can be controlled by means of differential flatness theory. This structural property enables the development of lightweight and cost-effective robotic systems with enhanced dexterity. However, a key challenge lies in managing the passive joints, whose control demands precise and comprehensive dynamic modeling of the system. To simplify dynamic models, particularly for low-speed trajectories, friction is often neglected. While this assumption simplifies analysis and control design, it introduces residual oscillations of the end-effector about the target position. In this paper, the possibility of using optimal control along with differential flatness control is investigated to improve the tracking of the planned trajectories. First, the study was carried out through formal analysis, and then, it was validated by means of numerical simulations. Results highlight that optimal control can be used to plan the flat variables considering different (quadratic) performance indices: control effort, i.e. motor torque, and potential energy of the considered underactuated joint. Moreover, the minimization of potential energy can be used to design motion laws that are robust against variation of the stiffness and damping of the underactuated joint, thus reducing oscillations in the case of stiffness/damping mismatch.

Figures & tables

Fig. 1: Scheme of the underactuated differentially flat robot with $n$ -DOF.

Parameter	Value	Unit
$I_{1}^{*}$	6.4 $\cdot 10^{-4}$	kgm 2
$I_{2}^{*}$	3.3 $\cdot 10^{-5}$	kgm 2
$k_{2}$	3 $\cdot 10^{-3}$	Nm/rad
$c_{2}$	2 $\cdot 10^{-5}$	Nms/rad

TABLE I: 2-DOF robot parameters.

Fig. 2: Angular positions of joint 1 (top) and 2 (middle) and joint 1 torque (bottom) obtained for scenario 1. The vertical solid line denotes the end of the motion.

Fig. 3: Angular positions of joint 1 (top) and 2 (middle) and joint 1 torque (bottom) obtained for scenario 2. The vertical solid line denotes the end of the motion.

Fig. 4: Angular positions of joint 1 (top) and 2 (middle) and joint 1 torque (bottom) obtained for scenario 3. The vertical solid line denotes the end of the motion.

Scenario	Strategy	Motor torque RMS (Nm)	Oscillation amplitude at $t=T$ (rad)
	Polynomial	0.0074	0
1	Min-control	0.0069 ( $-$ 6.8%)	0
	Min-potential	0.0087 ( $+$ 18%)	0
	Polynomial	0.0068	0.0045
2	Min-control	0.0064 ( $-$ 5.9%)	0.0036 ( $-$ 20%)
	Min-potential	0.0079 ( $+$ 16%)	0.003 ( $-$ 35%)

TABLE II: Results for the three simulated scenarios. The percentages refer to the Polynomial strategy of the same scenario.

Robotic Control

Robotic control studies how machines use sensing and computation to produce reliable movement and interaction in changing environments.

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Control Lyapunov Functions for Underactuated Soft Robots

Huy Pham, Zach J. Patterson

Mechanical and Aerospace Engineering, Case Western Reserve University

Robotic Control Soft Robotics Lyapunov-Based Control Robot Trajectory Tracking

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Soft and soft-rigid hybrid robots are inherently underactuated and operate under tight actuator limits, making task-space control with stability guarantees challenging. Common nonlinear strategies for soft robots (e.g., those based on PD control) often rely on the assumption of full actuation with no actuator limits. This paper presents a general control framework for task-space regulation and tracking of underactuated soft robots under bounded inputs. The method enforces a rapidly exponentially stabilizing control Lyapunov function as a convex inequality constraint while simultaneously satisfying underactuated full-body dynamics and actuator bounds. We validate the approach in simulation on several platforms spanning increasing underactuation: a simple two link tendon-driven "finger", a trimmed helicoid manipulator, and a highly underactuated spiral robot. We compare against a number of baseline methods from the literature. Results show improved task-space accuracy and consistent Lyapunov convergence under input limits, achieving superior set-point and trajectory-tracking performance.

Abstract

Robotic Control

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