Robot Dynamics

In my study of robot dynamics, I focus on understanding and modeling the motion of robotic systems. This involves analyzing the forces and torques that govern the movement of mechanisms and systems. My coursework and projects have centered around applying the Lagrangian formulation and Newton-Euler dynamics to multi-degree-of-freedom (DOF) systems, which is critical for designing and controlling advanced robotic mechanisms.

  • Lagrangian Dynamics: I have in-depth knowledge of applying the Lagrangian formulation to derive equations of motion for robotic systems. This method streamlines the process of analyzing the energy relationships in complex robotic mechanisms.

  • The Jacobian matrix is a critical tool in robot dynamics and control, as it relates joint velocities to the end-effector's linear and angular velocities. By understanding the Jacobian, I can analyze how forces and torques applied at the end-effector translate back to joint-level forces. This has been essential for solving problems in robotic manipulation, ensuring smooth and efficient motion planning.

  • Twists represent the velocity of a rigid body in space, combining angular velocity and linear velocity into a compact, six-dimensional vector. In robot dynamics, twists provide a unified framework for describing motion and are integral to understanding how different parts of a robotic system interact dynamically.

  • The Product of Exponentials (POE) formula is a powerful representation of forward kinematics for robotic manipulators. It simplifies the computation of a robot’s configuration by chaining exponentials of twist coordinates associated with each joint. This elegant mathematical framework allows me to model complex robotic structures efficiently, bridging kinematic and dynamic analyses.

  • Forward kinematics determines the position and orientation of a robot's end-effector given its joint parameters. It involves using the robot's geometric and kinematic model to compute the configuration of the end-effector in Cartesian space, enabling precise control for tasks like manipulation and navigation.

  • Inverse kinematics is the process of calculating the joint parameters required to achieve a desired end-effector position and orientation. This is often more complex than forward kinematics due to the potential for multiple solutions or constraints, but it is critical for achieving targeted motion in robotic systems.

  • The Newton-Euler method is a foundational approach in robotics for calculating the joint torques and forces needed to achieve a desired motion trajectory. By analyzing both translational and rotational dynamics, it recursively computes the forces and torques from the robot's end-effector to its base. This method is efficient and widely used in real-time control of robotic manipulators, enabling precise motion under various load and environmental conditions.

  • Rodrigues' rotation formula provides an efficient way to compute the rotation matrix for rotating a vector around an axis by a given angle. It simplifies the representation of 3D rotations using the axis-angle framework, avoiding the complexities of quaternions or Euler angles. This formulation is particularly useful in robotics for tasks involving rotational transformations, such as aligning end-effectors or planning motions in three-dimensional space.

RBE 501 - Final Project: Robot Dynamics

Overview

This project explores the dynamics and control of a robotic manipulator as part of the RBE 501 course. The objective is to apply theoretical knowledge and computational tools to solve practical robotics problems. The tasks include calculating transformations, implementing motion planning, and analyzing torque relationships. The final demonstration highlights the integration of these concepts into a functioning system.

Tasks

Task 1: Transformation and Joint Angles

  • Calculate the 4x4 transformation matrices for points A and B:
    • A = (185 mm, -185 mm, 185 mm)
    • B = (185 mm, 170 mm, 70 mm)
  • Determine the sets of joint angles (θ) for both positions with the gripper parallel to the table.
  • Analyze the uniqueness of the solutions.
  • Compare motor currents for Joint 2 at varying speeds (100%, 80%, 60%, 40%).

Task 2: Motion Planning with LSPB

  • Introduce an obstacle at a height of 200 mm between A and B.
  • Define a waypoint C at (185 mm, 0 mm, 240 mm).
  • Use Linear Segment with Parabolic Blend (LSPB) to transition from:
    • A → C → B (with a brief stop at C).
  • Visualize joint angle changes during a 20-second trajectory.

Task 3: Velocity-Based Control

  • Transition to velocity-based control for smooth motion between A, C, and B.
  • Maintain a constant velocity at C for a seamless transition.
  • Update velocity values dynamically to ensure precise path following over 20 seconds.

Task 4: Torque Estimation

  • Use current-based position control to hold the manipulator at A.
  • Derive torque estimates using current values and the InverseDynamics.m simulation from HW4.
  • Validate torque calculations with a known wrench applied at the end-effector.
  • Infer an unknown wrench based on torque estimates and compare with previous results.

Final Demonstration

The final demonstration consists of:

  1. Position-Based Control: Transition directly from A to B.
  2. Velocity-Based Control: Navigate the path A → C → B at constant velocity at C.
  3. Torque Estimation: Calculate the applied wrench using sensor data.

Report

  • Provide detailed explanations, equations, and solutions for all tasks.
  • Include graphs and figures illustrating the transformations, trajectories, and torque calculations.
  • Address challenges encountered and how they were resolved.

Requirements

  • Hardware: Robotic manipulator with current and velocity control capabilities.
  • Software: MATLAB (for InverseDynamics.m), simulation tools, and data visualization utilities.

Instructor

  • Course: RBE 501 - Robot Dynamics
  • Instructor: Andre Rosendo

Report and Presentation Deadline: Monday, December 11th

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