13.3 Mechanical systems and robotics

Mechanical systems and robotics apply dynamical systems principles to machines and robots. This topic explores how engineers use concepts like kinematics, dynamics, and control theory to design and operate complex mechanical systems.

From robot arms to autonomous vehicles, these principles enable precise movement and interaction with the environment. Understanding the math behind robot motion and control is crucial for developing advanced robotic systems in various industries.

Robot Kinematics

Degrees of Freedom and Manipulators

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• Degrees of freedom (DOF) refer to the number of independent parameters that define the configuration of a robotic system
  • Each DOF corresponds to a joint or axis of motion (rotational or translational)
  • Example: A robotic arm with 6 DOF can move in 6 different ways (3 rotational and 3 translational)
• Robot manipulators are composed of links connected by joints, forming a kinematic chain
  • Links are rigid bodies that connect the joints
  • Joints allow relative motion between the links (revolute joints for rotation, prismatic joints for translation)
  • End effector is the device at the end of the manipulator that interacts with the environment (gripper, tool, etc.)

Kinematics and Inverse Kinematics

• Kinematics is the study of motion without considering the forces that cause it
  • Forward kinematics determines the position and orientation of the end effector given the joint angles or positions
  • Inverse kinematics determines the joint angles or positions required to achieve a desired end effector position and orientation
• Denavit-Hartenberg (DH) convention is a systematic way to assign coordinate frames to the links of a robotic manipulator
  • DH parameters (link length, link twist, joint distance, joint angle) describe the geometric relationship between adjacent links
• Inverse kinematics is more complex than forward kinematics due to multiple solutions and singularities
  • Analytical methods solve the inverse kinematics equations directly (closed-form solutions)
  • Numerical methods iteratively search for a solution (Jacobian-based, optimization-based)
  • Example: Given a desired position and orientation of a robotic arm's end effector, inverse kinematics determines the joint angles needed to reach that configuration

Robot Dynamics and Control

Dynamics and Lagrangian Mechanics

• Dynamics is the study of motion considering the forces and torques that cause it
  • Forward dynamics determines the motion of the robot given the forces and torques applied to the joints
  • Inverse dynamics determines the forces and torques required to achieve a desired motion
• Lagrangian mechanics is a formulation of classical mechanics that uses generalized coordinates and energies
  • Lagrangian $L = T - V$, where $T$ is the kinetic energy and $V$ is the potential energy
  • Euler-Lagrange equation: $\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}}) - \frac{\partial L}{\partial q} = \tau$, where $q$ is the generalized coordinate and $\tau$ is the generalized force
• Equations of motion are derived using the Lagrangian formulation
  • Manipulator dynamics consider the inertial, Coriolis, centrifugal, gravitational, and friction forces
  • Example: The torque required to move a robotic arm depends on its mass, inertia, and the desired acceleration

Control Algorithms and Stability

• Control algorithms are used to make the robot follow a desired trajectory or maintain a desired position
  • PID (Proportional-Integral-Derivative) control is a widely used linear control technique
    • Proportional term reduces the error, integral term eliminates steady-state error, derivative term improves transient response
  • Computed torque control is a nonlinear control technique that uses the inverse dynamics model
    • Feedforward term compensates for the manipulator dynamics, feedback term stabilizes the system
• Stability is a critical aspect of robotic control systems
  • Lyapunov stability theory is used to analyze the stability of nonlinear systems
    • Lyapunov function $V(x)$ is a positive definite function that decreases along the system trajectories
    • System is stable if $\dot{V}(x) \leq 0$ and asymptotically stable if $\dot{V}(x) < 0$
• Adaptive control techniques adjust the controller parameters to handle uncertainties and variations in the system
  • Example: Model reference adaptive control (MRAC) adjusts the controller gains to make the system behave like a reference model

Motion Planning

Trajectory Planning

• Trajectory planning involves generating a feasible path for the robot to follow while satisfying constraints
  • Path planning finds a collision-free path from the start to the goal configuration
  • Trajectory planning adds timing information to the path, specifying the robot's position, velocity, and acceleration over time
• Interpolation methods generate smooth trajectories between waypoints
  • Linear interpolation connects waypoints with straight lines
  • Polynomial interpolation (cubic, quintic) generates smooth curves that pass through the waypoints
  • Spline interpolation (B-splines, NURBS) uses piecewise polynomial functions for local control and continuity
• Optimal control techniques find trajectories that minimize a cost function (time, energy, jerk)
  • Pontryagin's minimum principle provides necessary conditions for optimality
  • Dynamic programming solves the optimal control problem by breaking it down into smaller subproblems
  • Example: Minimum-jerk trajectory minimizes the integral of the squared jerk (rate of change of acceleration) to generate smooth and natural motions