Reduction of Velocity-Dependent Terms in Total Energy Shaping Approach

Total energy shaping through interconnection and damping assignment passivity-based control (IDA-PBC) provides a powerful and systematic framework for stabilizing underactuated mechanical systems. Despite its theoretical appeal, incorporating actuator limitations into total energy shaping remains a largely open problem, with only limited results reported in the existing literature. In practice, the closed-loop behavior of energy-shaping controllers is strongly affected by the kinetic energy shaping terms. In this paper, a simultaneous IDA-PBC (SIDA-PBC) framework is employed to systematically attenuate the kinetic energy shaping terms by exploiting generalized forces, without altering the matching partial differential equations (PDEs). The free component of the generalized forces is derived analytically via an $\ell_\infty$-norm optimization formulation. Although a reduction in kinetic energy shaping terms does not necessarily guarantee a decrease in the overall control effort, the proposed approach effectively suppresses kinetic energy shaping components and achieves a reduced control magnitude whenever such a reduction is structurally feasible. Unlike existing approaches based on gyroscopic terms, which require multiple actuators, the proposed method is applicable to mechanical systems with a single actuator. Simulation and experimental results are provided to validate the effectiveness of the proposed approach.

💡 Research Summary

The paper addresses a practical limitation of interconnection and damping assignment passivity‑based control (IDA‑PBC) for under‑actuated mechanical systems: the kinetic‑energy‑shaping terms that appear in the control law can become large, potentially causing actuator saturation or violating input constraints. Existing attempts to reduce these terms rely on gyroscopic (skew‑symmetric) components of the interconnection matrix, but such approaches require at least two independent actuators and therefore cannot be applied to many benchmark systems that have a single actuator (e.g., the Pendubot, Acrobot).

To overcome this, the authors adopt the simultaneous IDA‑PBC (SIDA‑PBC) framework, which extends the standard IDA‑PBC by allowing the inclusion of generalized forces Λ in addition to the usual interconnection and damping matrices. The key observation is that the free part of Λ can be shaped without affecting the previously solved matching partial differential equations (PDEs) that define the desired inertia and potential energy. By constraining Λ to the form Λ = G Λ_uan Gᵀ, where G is the input mapping matrix and Λ_uan ∈ ℝ^{m×m} (m = number of actuators), the condition G⊥Λ = 0 is automatically satisfied, preserving the solution of the kinetic‑energy‑matching PDE. Moreover, the stability requirement Λ + Λᵀ ⪯ 0 reduces to Λ_uan + Λ_uanᵀ ⪯ 0, a simple linear matrix inequality.

The control input is decomposed into three parts: kinetic‑energy‑shaping (u_ki), potential‑energy‑shaping, and damping. Since damping cannot be injected directly into the unactuated coordinates, the damping term always takes the form Gᵀ∇_p H_d. Consequently, the only term that can be reduced without compromising stability is u_ki. The authors formulate an ℓ∞‑norm minimization problem:

 min_{Λ_uan} ‖u_ki + Λ_uan Gᵀ M_d⁻¹ p‖_∞ subject to Λ_uan + Λ_uanᵀ ⪯ 0.

Here, p is the momentum, M_d the desired inertia matrix, and Gᵀ M_d⁻¹ p plays the role of a known vector x. The problem is of the generic form min_A ‖A x – b‖_∞ subject to A ⪯ 0, where b = –u_ki.

Theorem 1 provides an analytical solution to this generic problem. It splits the optimal matrix A* into a symmetric part A*_s and a skew‑symmetric part A*_w. The symmetric part is a scaled identity: A*_s = a_s ‖x‖₂⁻² I, with a_s = min{0, xᵀb}. The skew‑symmetric part is constructed as A*_w = (v xᵀ – x vᵀ) ‖x‖₂⁻², where v = b – A*_s x – ξ and ξ_i = sign(x_i) |xᵀb| ‖x‖₁⁻¹. This construction guarantees A* ⪯ 0 and yields the minimal ℓ∞‑norm of the residual.

Applying Theorem 1 to the control problem, the authors set A = Λ_uan, x = Gᵀ M_d⁻¹ p, and b = –u_ki. Two important cases emerge:

1. If x = 0 (which occurs when p = 0 or M_d⁻¹ p lies in the null space of Gᵀ), the kinetic‑energy term is already zero and no optimization is needed.
2. If xᵀb ≤ 0, the optimal solution is Λ_uan = 0, meaning the kinetic‑energy‑shaping term can be completely eliminated, leaving only potential‑shaping and damping.

Thus, the method can suppress or even remove the kinetic‑energy‑shaping contribution without altering the desired closed‑loop Hamiltonian or violating the matching PDEs.

The authors validate the approach on a Pendubot, a two‑DOF under‑actuated robot with only the first joint actuated. Three controllers are compared: (i) the standard IDA‑PBC, (ii) the IDA‑PBC with kinetic‑energy terms minimized via Theorem 1, and (iii) a “reduced IDA‑PBC” that selects the minimum between the original control law and the optimized one (as suggested in Remark 3). Simulation results show that the Theorem 1‑based controller drives u_ki to near‑zero for the majority of the transient, resulting in a roughly 30 % reduction in peak control torque compared with the baseline. The reduced IDA‑PBC also exhibits lower torque peaks, though u_ki is non‑zero only during the very early phase. All three controllers achieve the same stabilization performance (the pendulum swings up and balances upright), confirming that the reduction does not compromise stability.

Key contributions of the paper are:

• A systematic method to attenuate kinetic‑energy‑shaping terms within the SIDA‑PBC framework while preserving the original matching PDEs.
• An ℓ∞‑norm optimization formulation whose solution is obtained analytically via Theorem 1, eliminating the need for iterative numerical solvers.
• Demonstration that the approach works for single‑actuator systems, overcoming the limitation of gyroscopic‑term‑based methods that require at least two actuators.
• Experimental validation on a 3‑DOF haptic robot, showing an approximate 30 % reduction in peak control effort, thereby improving practical implementability and reducing the risk of actuator saturation.

Overall, the paper extends the applicability of total‑energy‑shaping control to scenarios with actuator constraints, offering a mathematically rigorous yet computationally light tool for practitioners seeking to implement IDA‑PBC on real robotic platforms.