Stable End-Effector Control (SEEC) Framework

• Stable End-Effector Control (SEEC) Framework is a model-based approach that enables stable hybrid position–force tracking in robotic manipulators by managing transitions between free and contact states.
• It employs a state-dependent switching control law with hybrid trajectory encoding to integrate motion and force references for precise manipulation in stiff environments.
• Incorporating a compliant wrist design reduces impractical damping demands, enhances impact mitigation, and improves tracking accuracy during contact transitions.

Stable End-Effector Control (SEEC) Framework refers to a class of model-based control architectures for robotic manipulators designed to guarantee stable and accurate tracking of hybrid position–force trajectories, especially during interaction with stiff environments. The definitive example of a SEEC framework is the switching position–force control law and system-level stability analysis detailed in (Heck et al., 2015). The framework systematically addresses the challenges inherent to switching dynamics arising when an end-effector transitions between free motion and constrained (contact) states, with an explicit focus on input-to-state stability (ISS), controller gain trade-offs, and manipulator design for impact mitigation.

1. Switching Control Law and Hybrid Trajectory Formulation

SEEC integrates a state-dependent switching law that enables the manipulator to follow pure motion references in free space and apply desired force profiles upon contact. The controller operates as follows:

• Free Motion (no contact, $x \leq 0$):

$$F_c = M_a\ddot{x}_d(t) + k_d\left(\dot{x}_d(t) - \dot{x}\right) + k_p\left(x_d(t) - x\right)$$

where $F_c$ is actuation force, and $x_d(t)$, $\dot{x}_d(t)$, and $\ddot{x}_d(t)$ are the desired motion profiles.

• Contact ( $x > 0$):

$$F_c = F_d(t) + k_f\left(F_d(t) - F_e\right) - b_f\dot{x}$$

with $F_d(t)$ the desired (time-varying) contact force, $F_e$ the measured environmental force, $k_f$ a force-error gain, and $b_f$ a contact-phase damping gain.

A critical innovation is encoding the force reference in a virtual desired position based on the (locally linear) Kelvin–Voigt environment model: $k_e x_d(t) + b_e \dot{x}_d(t) \approx F_d(t)$ where $k_e$, $b_e$ are the (possibly estimated) environment stiffness and damping parameters.

This "hybrid trajectory" approach unifies motion and force tracking by ensuring the end-effector’s state $x$ converges toward $x_d(t)$, and in contact, this also drives $F_e \to F_d(t)$.

2. Stability and Input-to-State Stability (ISS) Analysis of Switching Systems

The framework recasts the closed-loop system as a switched conewise-linear state–space model: $\dot{z} = A_i z + N w_i(t), \quad i \in \{1,2\}$ where $z = [x_d(t)-x,\; \dot{x}_d(t)-\dot{x}]^\top$. The matrices for each mode are:

• Free motion ($i=1$): $K_1 = k_p/M$, $B_1 = (k_d + b)/M$
• Contact ($i=2$): $K_2 = (1+k_f)k_e/M$, $B_2 = ((1+k_f)b_e+b_f+b)/M$

ISS is established by demonstrating GUAS (global uniform asymptotic stability) of the switched unperturbed system. Key sufficient conditions involve the spectral properties of $A_1, A_2$ and are summarized in GUES (globally uniformly exponentially stable) results. If a "visible" eigenvector exists for a negative eigenvalue in both modes, the system is GUAS. Absent that, the core criterion is: $\Lambda_1\Lambda_2 < 1$ where $\Lambda_i$ encode state transition scaling over each interval and are functions of the gain-to-eigenvalue ratios, with explicit formulas involving arctangent functions identified in the paper (Equations 11–13).

A key finding is that to ensure stability when tracking arbitrary hybrid trajectories—especially with a rigid manipulator—requires a very large $b_f$ (contact-phase damping), which is often impractical and leads to degraded tracking (sluggish response).

3. Manipulator Redesign with Compliant Wrist

To circumvent the impractical damping demand, SEEC advocates mechanical compliance at the wrist as a design strategy. The compliant wrist is modeled as a series connection: $$\begin{aligned} M\ddot{x} + b\dot{x} &= F_c - F_t \ M_t\ddot{x}_t &= F_t - F_e(x_t, \dot{x}_t) \ F_t &= k_t(x - x_t) + b_t(\dot{x} - \dot{x}_t) \end{aligned}$$ where $x$ and $x_t$ denote arm and tip positions, $M, M_t$ their respective masses, and $k_t, b_t$ the wrist's stiffness and damping.

Design guidelines specify:

• $M_t \ll M$ (light tip)
• $k_t \ll k_e$ (wrist much softer than environment)
• $b_t \gg b_e$ (wrist damping dominant)

The effective contact stiffness and damping as felt at the arm are reduced by: $$\bar{k}_e = k_t\frac{k_e}{k_t + k_e}, \qquad \bar{b}_e = b_t\frac{k_e}{k_t + k_e}$$ This dynamic compliance passively attenuates impact forces and allows the controller to achieve stability with practical values of $b_f$, maintaining high tracking performance and eliminating contact oscillations ("bouncing").

4. Simulation Results and Performance Insights

Simulations compare the rigid design to the compliant wrist variant:

• With small $b_f$ and a stiff manipulator, the arm tracks well in free motion but incurs large force overshoots and persistent bouncing upon contact.
• Increasing $b_f \to 9000 \ \mathrm{Ns/m}$ (for rigid design) achieves monotonic force tracking but at the cost of tracking speed; the system becomes underdamped and sluggish.
• Incorporating a compliant wrist (e.g., $$k_t = 5\times10^4 \ \mathrm{N/m},\ b_t > 170 \ \mathrm{Ns/m}$$) and using low $b_f$ demonstrates reduced impact force, fast settling, and accurate trajectory tracking. The reduced-order (quasi-static) model matches the system’s dominant behavior, validating model reduction as a basis for controller/tuner design.

5. Key Mathematical Constructs

The SEEC framework is grounded in these central mathematical formulas:

• Switching law:

$$F_c = M_a\ddot{x}_d(t) + k_d(\dot{x}_d(t) - \dot{x}) + k_p(x_d(t) - x), \quad x \leq 0$$

$$F_c = F_d(t) + k_f(F_d(t) - F_e) - b_f\dot{x}, \quad x > 0$$

• Hybrid trajectory encoding:

$\hat{k}_e x_d(t) + \hat{b}_e \dot{x}_d(t) = F_d(t)$

• Stability criteria:

$$\dot{z} = A_i z + N w_i(t), \qquad \Lambda_i = \left( \frac{K_i}{\omega_i} [\cdots]^{-1/2} \right)^{(-1)^i} \exp\Big( -\frac{B_i}{2\omega_i}\phi_i \Big)$$

Stability requires either visible eigenvector existence or $\Lambda_1\Lambda_2 < 1$.

6. Implications and Modularity

The SEEC framework fundamentally reveals the performance–stability trade-off in hybrid position–force tracking scenarios with switching dynamics. Excessive active damping required for stability under rigid actuation leads to degraded dynamic performance, while redesigning actuators for mechanical compliance offloads impact absorption to passive elements, decoupling stability from high active damping demands. Numerical results validate that, with compliance, stable, accurate tracking of arbitrary motion–force references is possible with physically realistic gains.

This paradigm is directly generalizable to manipulation tasks involving hybrid contact transitions and informs the mechanical design of precision arms intended for interaction with stiff or uncertain environments. The core principles—hybrid switching law, stability analysis of switched systems, and the role of compliant elements—constitute the basis for robust, stable end-effector control in modern robotics.