Force Control Applications

Overview

The value of force sensing is ultimately realized in control applications. This section introduces core applications of force sensing in robot control: force-controlled grasping, impedance control, admittance control, collision detection, and force-guided assembly.

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Force-Controlled Grasping

Grip Force Regulation

Goal: Apply just enough grip force -- neither dropping nor damaging the object.

Minimum grip force condition (parallel gripper, two-finger contact):

\[F_{grip} \geq \frac{F_{load}}{2\mu}\]

where:

• \(F_{load}\) is the external load (typically object weight \(mg\); during acceleration \(m(g + a)\))
• \(\mu\) is the friction coefficient between fingers and object
• Factor of 2 accounts for two contact surfaces

With safety factor:

\[F_{grip} = k_s \cdot \frac{m(g + a_{max})}{2\mu_{min}}\]

Typical safety factor \(k_s = 1.5 \sim 2.0\).

Slip Detection

When slipping occurs, the force sensor detects the following characteristic signals:

Based on tangential force ratio:

\[r = \frac{\sqrt{F_x^2 + F_y^2}}{F_z}\]

When \(r \to \mu\) (approaching the friction cone boundary), slip is imminent.

Based on vibration:

Slip generates high-frequency vibration signals (typically 50-500 Hz), detectable through the AC component of tactile sensors:

\[\text{slip\_indicator} = \int_{f_{low}}^{f_{high}} |FFT(F_{tangential}(t))|^2 df > \theta_{slip}\]

Adaptive Grip Control:

def adaptive_grip_control(F_sensor, F_grip_current):
    """
    Adaptive grip control based on slip detection
    """
    F_normal = F_sensor.z
    F_tangential = sqrt(F_sensor.x**2 + F_sensor.y**2)

    # Friction ratio
    friction_ratio = F_tangential / (F_normal + 1e-6)

    # Safety margin
    safety_margin = mu_estimated - friction_ratio

    if safety_margin < SAFETY_THRESHOLD:
        # Approaching slip, increase grip force
        F_grip_new = F_grip_current + GRIP_INCREMENT
    elif safety_margin > 2 * SAFETY_THRESHOLD:
        # Sufficient margin, can reduce grip force (save energy/protect object)
        F_grip_new = F_grip_current - GRIP_DECREMENT
    else:
        F_grip_new = F_grip_current

    return clip(F_grip_new, F_GRIP_MIN, F_GRIP_MAX)

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Impedance Control

Basic Concept

Impedance control makes the robot end-effector behave with desired "spring-damper-mass" dynamics:

\[F_{ext} = M_d \ddot{e} + D_d \dot{e} + K_d e\]

where:

• \(e = x - x_d\) is the position error (current position - desired position)
• \(M_d \in \mathbb{R}^{6 \times 6}\) is the desired inertia matrix
• \(D_d \in \mathbb{R}^{6 \times 6}\) is the desired damping matrix
• \(K_d \in \mathbb{R}^{6 \times 6}\) is the desired stiffness matrix
• \(F_{ext}\) is the external contact force

Physical Intuition

Parameter	Effect of Increasing	Typical Scenario
Large \(K_d\)	Robot is "stiff," resists displacement	Precision positioning
Small \(K_d\)	Robot is "soft," compliant to external force	Safe interaction
Large \(D_d\)	Motion is damped, slow response	Vibration suppression
Large \(M_d\)	Large inertia, hard to push	Heavy-load handling

Joint-Space Impedance Control

Mapping the impedance relationship to joint space:

\[\tau = J^T(q) F_{ext} + g(q) + M(q)\ddot{q}_d + C(q, \dot{q})\dot{q}_d\]

Control law:

\[\tau = g(q) + K_q(q_d - q) + D_q(\dot{q}_d - \dot{q})\]

where \(K_q, D_q\) are joint-space stiffness and damping matrices.

Cartesian-Space Impedance Control

More intuitive -- directly specifying impedance behavior in task space:

\[\tau = J^T(q) [K_d(x_d - x) + D_d(\dot{x}_d - \dot{x})] + g(q)\]

Important Notes:

• Requires accurate kinematic model (\(J(q)\), \(x = f(q)\))
• Dynamics compensation (gravity \(g(q)\)) improves performance
• Special handling needed near singularities

Variable Stiffness Control

Adjusting stiffness online according to task requirements:

\[K_d(t) = \begin{cases} K_{high} & \text{free-space motion} \\ K_{low} & \text{contact state} \end{cases}\]

Smooth transition:

\[K_d(t) = K_{low} + (K_{high} - K_{low}) \cdot \sigma(-\alpha \|F_{ext}\|)\]

where \(\sigma\) is the sigmoid function.

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Admittance Control

Difference from Impedance Control

Aspect	Impedance Control	Admittance Control
Input	Position deviation -> Output force	External force -> Output position deviation
Causality	\(F = Z \cdot v\) (Force = Impedance × Velocity)	\(v = Y \cdot F\) (Velocity = Admittance × Force)
Suitable For	Torque-controllable joints	Position-controlled robots
Sensor Requirement	Force sensor optional	Force sensor required
Inner Loop	Torque control	Position/Velocity control

Admittance Controller

The force sensor measures external force \(F_{ext}\), and the admittance controller computes position correction:

\[M_d \ddot{x}_{adj} + D_d \dot{x}_{adj} + K_d x_{adj} = F_{ext}\]

Then the correction is superimposed on the desired trajectory:

\[x_{cmd} = x_{desired} + x_{adj}\]

class AdmittanceController:
    def __init__(self, M, D, K, dt):
        self.M = M  # Desired inertia
        self.D = D  # Desired damping
        self.K = K  # Desired stiffness
        self.dt = dt
        self.x_adj = np.zeros(6)
        self.dx_adj = np.zeros(6)

    def update(self, F_ext, x_desired):
        # Admittance dynamics: M * ddx + D * dx + K * x = F_ext
        ddx_adj = np.linalg.solve(self.M,
            F_ext - self.D @ self.dx_adj - self.K @ self.x_adj)

        # Integration
        self.dx_adj += ddx_adj * self.dt
        self.x_adj += self.dx_adj * self.dt

        # Superimpose on desired position
        x_cmd = x_desired + self.x_adj
        return x_cmd

Application: Human-Robot Collaborative Carrying

A human pushes the robot end-effector, and the robot compliantly moves:

• \(K_d = 0\) (no spring restoring force, free movement)
• \(D_d\) moderate (damping provides stability and a "weight feel")
• \(M_d\) small (small inertia, easy to push)

\[D_d \dot{x}_{adj} = F_{ext} \quad \Rightarrow \quad \dot{x}_{adj} = \frac{F_{ext}}{D_d}\]

This is the simplest admittance control -- the human applies force, and the robot moves proportionally.

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Collision Detection

Momentum Observer

The momentum observer is a collision detection method that does not require an external force sensor (though having one improves accuracy).

Generalized momentum:

\[p = M(q)\dot{q}\]

Rate of momentum change:

\[\dot{p} = M(q)\ddot{q} + \dot{M}(q)\dot{q} = \tau + J^T F_{ext} - C(q,\dot{q})\dot{q} - g(q) + \dot{M}\dot{q}\]

Using the property that \(\dot{M} - 2C\) is skew-symmetric:

\[\dot{p} = \tau + J^T F_{ext} + C^T(q,\dot{q})\dot{q} - g(q)\]

Observer:

\[\hat{r}(t) = K_I \left[ p(t) - p(0) - \int_0^t (\tau + C^T \dot{q} - g + \hat{r}) d\tau' \right]\]

where \(K_I > 0\) is the observer gain. At steady state:

\[\hat{r} \to J^T F_{ext} = \hat{\tau}_{ext}\]

The observer output converges to the external torque.

Collision Judgment

\[\|\hat{\tau}_{ext}\| > \tau_{threshold} \quad \Rightarrow \quad \text{collision detected}\]

Collision Response Strategies:

Level	Condition	Response
Warning	\(\|\hat{\tau}_{ext}\| > \tau_1\)	Decelerate
Soft stop	\(\|\hat{\tau}_{ext}\| > \tau_2\)	Stop motion
Emergency stop	\(\|\hat{\tau}_{ext}\| > \tau_3\)	Cut motor drive
Retreat	After soft stop	Back away along collision direction

Improving Sensitivity

Collision detection sensitivity depends on:

1. Model accuracy: Inaccurate friction model -> large residual -> low sensitivity
2. Sensor noise: Encoder noise -> threshold cannot be set too low
3. Sampling rate: Higher is better (1 kHz or above recommended)
4. Filtering: Low-pass filtering reduces false alarms but increases latency

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Force-Guided Assembly

Peg-in-Hole Assembly

A classic force-controlled assembly task and the killer application for 6-axis F/T sensors.

Challenges:

• Tight clearance (possibly only 0.01-0.1 mm)
• Pure position control cannot accomplish this (positioning accuracy insufficient)
• Force feedback is needed to guide alignment

Three-Phase Strategy:

Phase 1: Approach and Search

Spiral search to locate the hole:

\[x(t) = x_0 + r(t)\cos(\omega t), \quad y(t) = y_0 + r(t)\sin(\omega t)\]

\[r(t) = r_0 + v_r \cdot t\]

Simultaneously apply constant downward force \(F_z = F_{search}\).

Detection condition: When \(F_z\) suddenly decreases (peg enters the hole opening) -> advance to next phase.

Phase 2: Alignment

Adjust orientation based on torque signals:

\[\Delta\theta_x = -K_\theta \cdot M_y, \quad \Delta\theta_y = K_\theta \cdot M_x\]

Torques \(M_x, M_y\) reflect the angular misalignment between peg and hole.

Convergence condition: \(\sqrt{M_x^2 + M_y^2} < M_{tolerance}\)

Phase 3: Insertion

Maintain orientation, apply constant force along the Z direction:

\[F_z = F_{insert} = \text{const}\]

Simultaneously zero lateral forces (admittance control):

\[\dot{x}_{adj} = \frac{1}{D} F_x, \quad \dot{y}_{adj} = \frac{1}{D} F_y\]

Completion condition: \(z > z_{target}\) or \(F_z > F_{max}\) (jammed).

Surface Tracking

Force sensors enable the robot to move along an unknown surface while maintaining constant contact force:

\[\text{Normal}: \quad F_n = F_{desired} \quad \text{(force control)}\]

\[\text{Tangential}: \quad v_t = v_{desired} \quad \text{(velocity control)}\]

Hybrid Force/Position Control:

\[\tau = J^T [S_f K_f (F_d - F) + S_p K_p (x_d - x)]\]

where:

• \(S_f = \text{diag}(0,0,1,0,0,0)\) -- Force control direction selection matrix
• \(S_p = I - S_f\) -- Position control direction selection matrix
• Normal direction (z) uses force control, tangential directions (x, y) use position control

Screw Tightening

A typical torque control application:

1. Free run-in phase: Velocity control, low torque
2. Seating phase: Torque begins to rise
3. Tightening phase: Torque control, reach target torque

\[M_z(t) \to M_{target} \quad \text{(e.g., 5 N·m)}\]

Over-tightening protection: \(M_z > M_{max} \Rightarrow \text{stop}\)

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Force Control + Learning

Learning Impedance Parameters

Traditional methods require manual tuning of \(K_d, D_d, M_d\); reinforcement learning can learn them automatically:

\[\pi(s) = \{x_d, K_d, D_d\}\]

State \(s\) includes position, velocity, and force information.

Residual Force Learning

Superimpose learned force corrections on top of model-based control:

\[F_{total} = F_{model}(q, \dot{q}, \ddot{q}) + F_{residual}^{NN}(q, \dot{q}, F_{ext})\]

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Related Content

• Control Theory -- Control theory fundamentals
• 6-Axis Force/Torque Sensor -- F/T sensor hardware
• Force Sensing Overview -- Complete picture of force sensing

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References

• Siciliano, B. et al., Robotics: Modelling, Planning and Control, Ch. 9
• Hogan, N., "Impedance Control: An Approach to Manipulation," 1985
• De Luca, A. et al., "Collision Detection and Safe Reaction with the DLR-III Lightweight Manipulator Arm," IROS, 2006
• Inoue, T. et al., "Robotic Peg-in-Hole Assembly: State of the Art Review," Advanced Robotics, 2017

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