1.2 Robot dynamics

Robot dynamics forms the backbone of understanding and controlling robotic systems. It combines principles from physics, mathematics, and engineering to predict and influence robot behavior under various conditions and forces.

This topic covers essential concepts like Newton-Euler and Lagrangian formulations, kinematic chains, and forward and inverse dynamics. It also delves into rigid body dynamics, multi-body systems, and energy-based methods, providing a comprehensive framework for analyzing and designing robotic systems.

Fundamentals of robot dynamics

• Provides essential framework for understanding and predicting robot motion in Robotics and Bioinspired Systems
• Encompasses various mathematical approaches to describe robot behavior under different forces and conditions
• Forms the foundation for advanced robot control, planning, and design in biomimetic systems

Newton-Euler formulation

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• Utilizes Newton's laws of motion and Euler's equations for rotational dynamics
• Applies to each link of the robot separately, considering forces and torques
• Involves recursive algorithms for efficient computation of joint torques
• Suitable for real-time control applications due to its computational speed
• Requires knowledge of acceleration, velocity, and position of each link

Lagrangian formulation

• Based on the principle of least action and energy conservation
• Describes robot dynamics using generalized coordinates and energies
• Eliminates constraint forces, simplifying the analysis of complex systems
• Produces a set of second-order differential equations of motion
• Particularly useful for systems with many degrees of freedom
• Facilitates the derivation of closed-form dynamic equations

Principle of virtual work

• States that the sum of virtual work done by all forces is zero for equilibrium
• Applies to both static and dynamic systems in robotics
• Useful for analyzing systems with complex constraints or redundant degrees of freedom
• Helps in deriving equations of motion for robotic systems
• Can be extended to include the concept of virtual power for dynamic analysis

Kinematic chains

• Fundamental concept in robot structure and motion analysis for Robotics and Bioinspired Systems
• Describes the arrangement and connectivity of robot links and joints
• Crucial for understanding robot workspace, manipulability, and dynamic properties

Open vs closed chains

• Open chains have one fixed end and one free end (robotic arms)
• Closed chains form loops with no free ends (parallel robots, walking mechanisms)
• Open chains offer larger workspace and simpler kinematics
• Closed chains provide higher stiffness and load-bearing capacity
• Hybrid systems combine both types for specific applications (humanoid robots)

Degrees of freedom

• Represent the number of independent variables needed to describe the robot's configuration
• Determined by the number and type of joints in the kinematic chain
• Affects the robot's mobility, dexterity, and control complexity
• Can be calculated using Grübler's formula for planar and spatial mechanisms
• May be redundant or deficient depending on the task requirements

Joint types and constraints

• Revolute joints allow rotation about a single axis
• Prismatic joints permit linear motion along one axis
• Spherical joints enable rotation about three orthogonal axes
• Universal joints combine two revolute joints with perpendicular axes
• Constraints limit joint motion (mechanical stops, singularities)
• Selection of joint types influences robot kinematics, dynamics, and control strategies

Forward dynamics

• Predicts robot motion given applied forces and torques in Robotics and Bioinspired Systems
• Essential for simulation, motion planning, and control system design
• Involves solving complex differential equations to determine robot behavior

Equations of motion

• Derived from Newton-Euler or Lagrangian formulations
• Express accelerations as functions of joint positions, velocities, and applied forces/torques
• Include mass matrix, Coriolis and centrifugal terms, and gravity effects
• Can be written in the general form: $M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) = \tau$
• M(q) represents the mass matrix, C(q,ẋ) the Coriolis and centrifugal terms, G(q) the gravity vector, and τ the applied joint torques

Numerical integration methods

• Used to solve equations of motion over time
• Euler method provides simple but less accurate solutions
• Runge-Kutta methods offer higher accuracy at increased computational cost
• Implicit methods handle stiff systems more effectively
• Adaptive step-size algorithms balance accuracy and computational efficiency

Simulation techniques

• Time-stepping approaches discretize the continuous dynamics
• Event-driven methods handle discontinuities (impacts, friction transitions)
• Multi-rate integration techniques simulate systems with different time scales
• Co-simulation combines different solvers for subsystems (rigid bodies, flexible elements)
• Hardware-in-the-loop simulations incorporate physical components with virtual models

Inverse dynamics

• Calculates required forces and torques to achieve desired robot motion in Robotics and Bioinspired Systems
• Critical for feedforward control, trajectory optimization, and robot design
• Involves solving equations of motion in reverse to determine input forces

Torque calculation

• Computes joint torques needed to produce specified joint accelerations, velocities, and positions
• Utilizes the robot's dynamic model and desired trajectory information
• Can be performed using recursive Newton-Euler algorithm or Lagrangian formulation
• Accounts for gravitational, inertial, and Coriolis/centrifugal effects
• Requires accurate knowledge of robot parameters (masses, inertias, link lengths)

Force estimation

• Determines end-effector forces and moments from joint torques and kinematics
• Useful for interaction control and force feedback in robotic manipulation
• Involves solving the Jacobian transpose equation: $\tau = J^T(q)F$
• Can be extended to estimate internal forces in closed-chain mechanisms
• Considers friction and other non-ideal effects for accurate estimation

Applications in control

• Feedforward control compensates for known dynamic effects
• Computed torque control linearizes robot dynamics for improved tracking
• Model predictive control optimizes future trajectories based on dynamic predictions
• Adaptive control adjusts dynamic parameters online to improve performance
• Impedance control regulates the relationship between force and motion

Rigid body dynamics

• Fundamental concept in robot dynamics for Robotics and Bioinspired Systems
• Assumes robot links are non-deformable bodies with fixed mass properties
• Forms the basis for more complex dynamic models and control strategies

Inertia tensors

• 3x3 matrices describing mass distribution of rigid bodies
• Defined relative to the body's center of mass or a reference frame
• Diagonal elements represent moments of inertia about principal axes
• Off-diagonal elements indicate products of inertia
• Transform between different coordinate frames using the parallel axis theorem
• Key component in calculating rotational dynamics of robot links

Angular momentum

• Conserved quantity in the absence of external torques
• Calculated as the product of inertia tensor and angular velocity
• Plays crucial role in attitude control of space robots and flying robots
• Relates to the gyroscopic effects in rotating machinery and robot joints
• Can be used to analyze and design balanced mechanisms

Coriolis and centrifugal forces

• Arise from the interaction of multiple moving bodies or reference frames
• Coriolis force is proportional to the cross product of angular velocity and linear velocity
• Centrifugal force acts radially outward from the axis of rotation
• Significant in high-speed robot operations and space robotics
• Represented in robot dynamics equations by the term C(q,ẋ)ẋ
• Can be computed efficiently using the Christoffel symbols of the first kind

Multi-body dynamics

• Extends rigid body dynamics to systems of interconnected bodies in Robotics and Bioinspired Systems
• Essential for analyzing complex robotic systems (humanoids, multi-limbed robots)
• Enables efficient simulation and control of robots with many degrees of freedom

Recursive algorithms

• Exploit the tree-like structure of open-chain robots for efficient computation
• Include forward recursion for kinematics and backward recursion for forces/torques
• Articulated Body Algorithm (ABA) computes forward dynamics in O(n) time
• Composite Rigid Body Algorithm (CRBA) efficiently assembles the mass matrix
• Recursive Newton-Euler Algorithm (RNEA) calculates inverse dynamics rapidly

Computational efficiency

• Sparse matrix techniques leverage the structure of robot dynamics equations
• Symbolic computation generates optimized code for specific robot configurations
• Parallel computation distributes calculations across multiple processors
• Model order reduction techniques simplify complex models for real-time control
• Hierarchical approaches decompose the dynamics of complex systems into subsystems

Parallel computation methods

• Divide-and-conquer algorithms split the kinematic chain for parallel processing
• GPU acceleration leverages graphics hardware for matrix operations
• Domain decomposition methods partition the robot into subdomains for parallel solving
• Warp-synchronous programming optimizes thread utilization on GPUs
• Load balancing techniques ensure efficient distribution of computational tasks

Dynamic parameter identification

• Process of determining accurate dynamic parameters for robot models in Robotics and Bioinspired Systems
• Critical for achieving high-performance control and simulation accuracy
• Combines theoretical modeling with experimental data to refine robot models

Least squares estimation

• Formulates parameter identification as an optimization problem
• Minimizes the sum of squared errors between measured and predicted dynamics
• Can handle overdetermined systems with more measurements than parameters
• Requires careful excitation of robot dynamics to ensure parameter identifiability
• May use regularization techniques to handle ill-conditioned problems
• Can be extended to weighted least squares for handling measurement uncertainties

Adaptive techniques

• Update dynamic parameters online during robot operation
• Include methods like Model Reference Adaptive Control (MRAC) and adaptive observers
• Can compensate for time-varying parameters (payload changes, wear)
• Often combine parameter estimation with adaptive control laws
• May use persistence of excitation conditions to ensure parameter convergence

Experimental methods

• Involve collecting data from robot motions to estimate dynamic parameters
• Include static tests for gravity compensation and friction identification
• Dynamic tests excite robot through various trajectories to capture inertial properties
• May use specialized equipment (force/torque sensors, accelerometers) for data collection
• Require careful design of excitation trajectories to ensure parameter identifiability
• Often employ data filtering and signal processing techniques to improve estimation accuracy

Energy-based methods

• Utilize energy principles to analyze and control robot dynamics in Robotics and Bioinspired Systems
• Provide insights into system stability and efficiency
• Facilitate the design of energy-optimal trajectories and control strategies

Hamiltonian mechanics

• Reformulates dynamics in terms of generalized coordinates and momenta
• Uses the Hamiltonian function, representing total energy of the system
• Leads to first-order equations of motion, simplifying certain analyses
• Facilitates the study of conserved quantities and symmetries in robot dynamics
• Useful for designing energy-shaping controllers and analyzing underactuated systems

Conservation principles

• Apply fundamental physical laws to robot dynamics (energy, momentum)
• Used to derive constraints and simplify dynamic equations
• Enable the analysis of passive dynamic walking and energy-efficient locomotion
• Facilitate the design of balanced mechanisms and energy-recuperation systems
• Can be exploited to develop robust control strategies for uncertain environments

Energy optimization

• Aims to minimize energy consumption in robot motions and tasks
• Involves formulating optimal control problems with energy-related cost functions
• Can lead to smooth and natural-looking trajectories in humanoid robotics
• Considers regenerative braking and energy storage in electric robot actuators
• Applies to battery life extension in mobile robots and long-term autonomy

Dynamics in constrained systems

• Addresses robot behavior when interacting with environment in Robotics and Bioinspired Systems
• Critical for tasks involving manipulation, locomotion, and assembly
• Requires integration of contact models with robot dynamics

Contact modeling

• Describes the interaction forces between robot and environment
• Includes compliant models (spring-damper) and rigid body contact models
• Considers normal forces, tangential forces, and impact dynamics
• May use penalty methods or constraint-based approaches for simulation
• Incorporates hysteresis effects for more realistic contact behavior
• Crucial for accurate simulation of grasping and locomotion tasks

Friction effects

• Model the resistance to relative motion between contacting surfaces
• Include static friction, kinetic friction, and viscous friction models
• Coulomb friction model widely used for its simplicity and effectiveness
• More advanced models consider stick-slip phenomena and Stribeck effect
• Significant impact on robot control, especially in precision manipulation tasks
• Can be compensated for using friction observers and adaptive control techniques

Impact dynamics

• Describes the rapid change in velocities and forces during collisions
• Involves impulse-momentum equations for instantaneous impacts
• Considers coefficient of restitution for energy loss during collisions
• May use compliant contact models for finite-duration impacts
• Critical for analyzing and controlling robot locomotion and manipulation
• Requires careful handling in numerical simulations to ensure stability

Robot dynamics software

• Essential tools for analysis, simulation, and control in Robotics and Bioinspired Systems
• Facilitate rapid prototyping and testing of robot designs and control algorithms
• Range from general-purpose packages to specialized robotics software

Simulation tools

• Include physics engines for realistic dynamic simulation (ODE, Bullet, MuJoCo)
• Provide 3D visualization and sensor simulation capabilities
• Often support co-simulation with control software and hardware-in-the-loop testing
• May include libraries for common robot models and components
• Examples: Gazebo, V-REP, Webots, ADAMS, MATLAB Simscape Multibody

Analysis packages

• Offer symbolic and numerical tools for deriving and solving dynamic equations
• Provide libraries for kinematics, dynamics, and trajectory optimization
• Support parameter identification and model validation
• Often integrate with simulation tools for comprehensive robot analysis
• Examples: MATLAB Robotics Toolbox, SymPy, Pinocchio, RBDL, Drake

Real-time implementations

• Focus on efficient computation of dynamics for real-time control
• Utilize optimized algorithms and code generation for specific robot configurations
• May leverage parallel computing and GPU acceleration for complex systems
• Often provide interfaces to common robot hardware and control architectures
• Examples: OROCOS, ROS Control, EtherCAT-based control frameworks

Advanced topics

• Explores cutting-edge areas in robot dynamics for Robotics and Bioinspired Systems
• Addresses challenges in specialized robotics applications and environments
• Combines principles from multiple disciplines for innovative solutions

Flexible body dynamics

• Considers deformation of robot links and joints under loading
• Utilizes finite element methods or assumed modes for modeling flexibility
• Important for lightweight, high-speed, and large-scale robotic systems
• Addresses vibration suppression and precision control in flexible manipulators
• Combines structural dynamics with rigid body dynamics for comprehensive modeling
• Applications in space robotics, large-scale manufacturing, and soft robotics

Underwater robot dynamics

• Accounts for hydrodynamic effects (added mass, drag, buoyancy)
• Considers fluid-structure interaction in robot design and control
• Addresses challenges of communication and localization in aquatic environments
• Incorporates models for thrusters and propulsion systems
• Applications in ocean exploration, underwater manipulation, and marine robotics
• Requires consideration of pressure effects and water-proofing in robot design

Space robot dynamics

• Deals with zero-gravity or micro-gravity environments
• Addresses challenges of momentum management and attitude control
• Considers the impact of orbital mechanics on robot motion
• Incorporates models for reaction wheels, thrusters, and tethers
• Applications in satellite servicing, space station maintenance, and planetary exploration
• Requires consideration of thermal effects and radiation in robot design and operation