7.4 Adaptive control

Adaptive control is a game-changer for soft robotics. It allows robots to adjust their behavior on the fly, adapting to changes in their own structure or environment. This flexibility is crucial for soft robots, which can deform and interact with their surroundings in complex ways.

The main types of adaptive control are Model Reference Adaptive Control and Self-Tuning Regulators. These techniques use parameter estimation and stability analysis to ensure robots perform well despite uncertainties. Applications include adaptive grasping, locomotion, and compliance control in soft robotics.

Adaptive control fundamentals

• Adaptive control enables systems to automatically adjust their control parameters in response to changes in the system or environment, making it well-suited for soft robotics applications where the robot's dynamics may vary due to deformation or interaction with the environment
• Adaptive control techniques can be classified into two main categories: model reference adaptive control (MRAC) and self-tuning regulators (STR), both of which aim to achieve desired performance in the presence of uncertainties or variations in the system

Model reference adaptive control

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• MRAC involves designing a reference model that represents the desired closed-loop behavior of the system and adjusting the controller parameters to minimize the error between the actual system output and the reference model output
• The adaptation mechanism in MRAC typically consists of an adaptation law that updates the controller parameters based on the error signal and a stability proof using Lyapunov theory to ensure boundedness of the error and parameter estimates
• MRAC has been applied to soft robotics for tasks such as trajectory tracking control of soft manipulators (pneumatic actuators) and force control of soft grippers (electroactive polymers)

Self-tuning regulators

• STR is an adaptive control approach that estimates the unknown system parameters online and updates the controller parameters based on these estimates to maintain desired performance
• The parameter estimation in STR can be performed using techniques such as recursive least squares (RLS) or gradient descent methods, which are discussed in the next section
• STR has been employed in soft robotics for applications like adaptive impedance control of soft exoskeletons (series elastic actuators) and adaptive vibration control of soft structures (piezoelectric actuators)

Parameter estimation techniques

• Parameter estimation is a crucial component of adaptive control, as it enables the controller to learn and adapt to the unknown or time-varying system parameters
• Two commonly used parameter estimation techniques in adaptive control are recursive least squares (RLS) and gradient descent methods, both of which aim to minimize the error between the estimated and actual system parameters

Recursive least squares

• RLS is an online parameter estimation method that recursively updates the parameter estimates by minimizing the weighted sum of squared errors between the predicted and measured system outputs
• The RLS algorithm consists of two main steps: 1) computing the parameter update based on the current estimation error and the covariance matrix, and 2) updating the covariance matrix using the matrix inversion lemma
• RLS has fast convergence properties and can handle time-varying parameters, making it suitable for real-time applications in soft robotics (adaptive control of soft manipulators)

Gradient descent methods

• Gradient descent is an iterative optimization algorithm that updates the parameter estimates in the direction of the negative gradient of the cost function, which represents the estimation error
• The learning rate in gradient descent determines the step size of the parameter update and can be fixed or adaptive (e.g., decreasing over time or using line search methods)
• Gradient descent methods, such as stochastic gradient descent (SGD) and its variants (Adam, RMSprop), are computationally efficient and can handle large-scale problems, making them applicable to learning-based control of soft robots (reinforcement learning)

Stability analysis

• Stability analysis is essential in adaptive control to ensure that the closed-loop system remains stable and converges to the desired performance in the presence of uncertainties and parameter adaptation
• Two main approaches for stability analysis in adaptive control are Lyapunov stability theory and robust adaptive control, both of which provide sufficient conditions for guaranteeing the stability and convergence of the adaptive system

Lyapunov stability theory

• Lyapunov stability theory is a powerful tool for analyzing the stability of nonlinear systems, including adaptive control systems
• The main idea behind Lyapunov stability is to find a positive definite function (Lyapunov function) that decreases along the system trajectories, implying that the system converges to an equilibrium point
• In adaptive control, the Lyapunov function typically includes terms related to the tracking error and the parameter estimation error, and the stability proof involves showing that the derivative of the Lyapunov function is negative definite or semi-definite

Robust adaptive control

• Robust adaptive control aims to design adaptive controllers that maintain stability and performance in the presence of uncertainties, disturbances, and unmodeled dynamics
• Robust adaptive control techniques include: 1) σ-modification, which adds a damping term to the adaptation law to prevent parameter drift, 2) projection operator, which constrains the parameter estimates to a known bounded set, and 3) dead-zone modification, which stops adaptation when the tracking error is within a specified threshold
• These modifications help to improve the robustness of the adaptive controller and have been applied to soft robotics applications such as robust tracking control of soft continuum manipulators (cable-driven actuation) and robust force control of soft grippers (jamming transition)

Applications in soft robotics

• Adaptive control has found numerous applications in soft robotics, where the inherent compliance and nonlinear dynamics of soft materials pose challenges for traditional control approaches
• Some key application areas of adaptive control in soft robotics include adaptive grasping and manipulation, adaptive locomotion control, and adaptive compliance control

Adaptive grasping and manipulation

• Soft grippers and manipulators can adapt their shape and stiffness to conform to objects of various geometries and properties, enabling versatile and delicate grasping and manipulation tasks
• Adaptive control can be used to automatically adjust the grasping force and stiffness based on the object's properties (fragility, weight) and the task requirements (precision, speed)
• Examples include adaptive impedance control of soft pneumatic grippers for handling delicate objects (fruits, eggs) and adaptive shape control of soft manipulators for reaching in cluttered environments (pipes, tubes)

Adaptive locomotion control

• Soft robots can achieve diverse modes of locomotion, such as crawling, rolling, and swimming, by exploiting their body compliance and interaction with the environment
• Adaptive control can help soft robots to automatically adapt their gait patterns and locomotion parameters (frequency, amplitude) based on the terrain conditions (stiffness, friction) and the robot's state (posture, velocity)
• Applications include adaptive gait learning for soft crawling robots (caterpillar-inspired) and adaptive swimming control for soft underwater robots (jellyfish-inspired)

Adaptive compliance control

• Soft robots can actively modulate their compliance to achieve desired interaction behaviors with the environment or the user, such as assisting human motion or absorbing impact forces
• Adaptive control can be employed to automatically adjust the compliance of soft robots based on the interaction forces and the task requirements (safety, efficiency)
• Examples include adaptive impedance control of soft exoskeletons for human-robot interaction (rehabilitation, assistance) and adaptive stiffness control of soft landing gear for adaptive shock absorption (drones, rovers)

Challenges and limitations

• Despite the promising potential of adaptive control in soft robotics, there are several challenges and limitations that need to be addressed for practical implementation and further advancement of the field
• Some key challenges and limitations include the trade-off between convergence speed and robustness, computational complexity, and sensor requirements and noise

Convergence speed vs robustness

• Adaptive control algorithms often face a trade-off between convergence speed and robustness, as faster adaptation may lead to higher sensitivity to noise and disturbances
• Techniques such as σ-modification and dead-zone modification can improve robustness but may slow down the convergence rate of the adaptive controller
• Balancing this trade-off requires careful tuning of the adaptation gains and modification parameters based on the specific application requirements and operating conditions

Computational complexity

• Adaptive control algorithms, especially those involving online parameter estimation and optimization, can be computationally demanding and may require significant memory and processing power
• This computational complexity can limit the real-time implementation of adaptive control on resource-constrained platforms, such as embedded systems and microcontrollers
• Strategies to mitigate this issue include using simplified models, offline learning, and hardware acceleration (FPGA, GPU)

Sensor requirements and noise

• Adaptive control relies on accurate and timely measurements of the system states and outputs for parameter estimation and control adaptation
• Soft robots often require specialized sensors (stretch, pressure, force) that can be integrated into the soft material without compromising its compliance and durability
• Sensor noise, drift, and delays can degrade the performance of the adaptive controller and may require additional filtering, calibration, and compensation techniques (Kalman filter, sensor fusion)