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
I1∗
6.4 ⋅10−4
kgm 2
I2∗
3.3 ⋅10−5
kgm 2
k2
3 ⋅10−3
Nm/rad
c2
2 ⋅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.
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.
Huy Pham, Zach J. Patterson
Mechanical and Aerospace Engineering, Case Western Reserve University
An asymmetric two-link robot supported atop a flat platform by wheels that roll and pivot freely, but do not slip laterally, will develop forward momentum if the joint between the links is actuated internally. In particular, oscillations in the joint angle will generate undulatory locomotion suggesting fishlike swimming. If two such robots surmount a common platform that's free to translate with its own inertial dynamics, then the individual robots' dynamics will be coupled so that the locomotion of either robot is affected by that of the other. We develop a mathematical model for this system and present simulations demonstrating its behavior. We then consider a single robot with an unactuated joint rolling atop a platform that moves under control, and show that actuation of the platform is sufficient to dictate the robot's behavior. In particular, with the acceleration of the platform as an input, the robot's heading can be made to track a chosen function of time. This is sufficient to guarantee that the robot can be induced to orbit a fixed point on the platform or to locomote persistently in a desired direction.
Hamidreza Moradi, Scott David Kelly
Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte, Charlotte, NC 28223 USA
Learning-based control techniques use data from past trajectories to control systems with uncertain dynamics. However, learning-based controllers are often computationally inefficient, limiting their practicality. To address this limitation, we propose a learning-based controller that exploits differential flatness, a property of many robotic systems. Recent research on using flatness for learning-based control either is limited in that it (i) ignores input constraints, (ii) applies only to single-input systems, or (iii) is tailored to specific platforms. In contrast, our approach uses a system extension and block-diagonal cost formulation to control general multi-input, nonlinear, affine systems. Furthermore, it satisfies input and half-space flat state constraints and guarantees probabilistic Lyapunov decrease using only two sequential convex optimizations. We show that our approach performs similarly to, but is multiple times more efficient than, a Gaussian process model predictive controller in simulation, and achieves competitive tracking in real hardware experiments.
Tobias A. Farger, Adam W. Hall, Angela P. Schoellig
Learning Systems and Robotics Lab at the Technical University of Munich, Munich, Germany. · Learning Systems and Robotics Lab at the University of Toronto Institute for Aerospace Studies (UTIAS) and the Vector Institute for Artificial Intelligence, Toronto, Canada