2.4 Velocity kinematics and static forces
2 min read•july 25, 2024
The is a powerful tool in robot kinematics, mapping to . It's crucial for understanding how a robot's joints affect its overall motion and for solving forward and inverse velocity problems.
Static forces and manipulability are key concepts in robot control. The relates end-effector forces to , while manipulability measures help assess a robot's dexterity and performance in different configurations.
Jacobian Matrix and Robot Kinematics
Jacobian matrix for robot manipulators
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- Jacobian matrix linearly maps joint velocities to end-effector velocities representing instantaneous kinematics of the robot
- Geometric approach considers linear and angular velocity contributions of each joint (revolute, prismatic)
- Analytical approach uses partial derivatives of equations with respect to joint variables
- (top 3 rows) and (bottom 3 rows) form complete Jacobian
- Dimensionality depends on number of (DOF) and (6x6 for 6-DOF robot)
Joint vs end-effector velocities
- : relates joint and end-effector velocities
- Forward velocity kinematics calculates end-effector velocities from given joint velocities
- Inverse velocity kinematics determines required joint velocities for desired end-effector velocities
- used for redundant manipulators with more DOF than task space dimensions
- Singularities occur when Jacobian loses rank impacting velocity relationships (, )
Static Forces and Manipulability
Static forces via Jacobian transpose
- : relates end-effector forces to joint torques
- equates work done in joint space to work in task space
- determines joint torques to counteract external forces ()
- visualizes in different directions (major axis indicates direction of maximum force)
Jacobian and robot manipulability
- indicates dexterity of manipulator at given configuration
- (ratio of max to min singular values) shows isotropy of manipulator configuration
- graphically represents velocity or force transmission capabilities (shape indicates dexterity)
- Impacts robot performance through , , and singularity avoidance (optimizing trajectories)
Key Terms to Review (28)
Angular Velocity Jacobian: The Angular Velocity Jacobian is a mathematical matrix that relates the joint velocities of a robotic manipulator to the end-effector's angular velocity. It plays a crucial role in velocity kinematics by transforming joint space velocities into task space velocities, allowing for a comprehensive understanding of how motion at the joints influences the motion of the end effector. This concept is essential for analyzing static forces and ensuring accurate control of robotic movements.
Condition Number of Jacobian: The condition number of the Jacobian is a numerical value that measures the sensitivity of a system of equations to changes in input values. It provides insight into how small variations in the input can lead to large changes in the output, highlighting potential issues with stability and performance in robotic systems. A high condition number indicates that the system is close to singularity, which may result in inaccuracies during calculations, especially when dealing with velocity kinematics and workspace analysis.
Degrees of Freedom: Degrees of freedom refer to the number of independent parameters or movements that a mechanical system can undergo. In robotics, it specifically indicates how many distinct ways a manipulator can move, which affects its ability to position and orient its end effector in space. Understanding degrees of freedom is essential for tasks like inverse kinematics, controlling velocities, assessing manipulator structures, and navigating configuration spaces with obstacles.
Elbow singularity: An elbow singularity occurs in robotic arms with a joint configuration resembling a human elbow when the arm reaches a position where it can no longer determine a unique orientation for the end effector. This situation arises due to the alignment of certain joints, leading to a loss of degrees of freedom and creating challenges in controlling the arm's movement and positioning. Understanding elbow singularities is crucial for effective velocity kinematics and analyzing static forces acting on the robotic system.
End-effector velocities: End-effector velocities refer to the speed and direction at which the end-effector of a robotic manipulator moves in space. This concept is crucial for understanding how robotic systems interact with their environment, as it involves the translation and rotation of the end-effector during operation. Mastering end-effector velocities is essential for ensuring accurate positioning and effective task performance in robotics, particularly in tasks that require precision and coordination.
Force Ellipsoid: The force ellipsoid is a geometric representation that illustrates the relationship between the forces exerted on a robotic end-effector and the resulting motion or acceleration of that end-effector. It provides insight into how various external forces can affect the control and manipulation of robotic systems, emphasizing the importance of understanding static forces in achieving precise movements.
Force Transformation Equation: The force transformation equation describes how forces are related and transformed between different coordinate systems, especially in the context of robotics and dynamics. It enables the calculation of forces acting on a robotic system by accounting for the geometry and orientation of the system, making it essential for analyzing both kinematic motion and static equilibrium.
Force Transmission Capabilities: Force transmission capabilities refer to the ability of a mechanical system to effectively transfer forces and moments through its structure during motion or when subjected to external loads. This concept is essential in understanding how different components of a robotic system interact and contribute to its overall performance, especially in relation to motion and stability under various loading conditions.
Forward Kinematics: Forward kinematics is the process of calculating the position and orientation of a robot's end effector based on the joint parameters, such as angles and displacements. This process is crucial for understanding how movements in a robotic system relate to its physical configuration, enabling precise control and manipulation in various applications.
Gravity Compensation: Gravity compensation refers to the technique used in robotics to counteract the effects of gravitational forces acting on a robotic system. By adjusting the control inputs to the actuators, gravity compensation ensures that the robot can maintain its position and perform tasks without being overly influenced by its weight. This concept is closely tied to understanding velocity kinematics and static forces, as it plays a vital role in achieving smooth and controlled movements while minimizing energy consumption.
Inverse Kinematics: Inverse kinematics is the process of calculating the joint parameters needed to place the end-effector of a robotic arm or manipulator at a desired position and orientation in space. This technique is essential for controlling robotic systems, as it allows for precise movement and positioning based on the goals set by a user or program.
Jacobian Matrix: The Jacobian matrix is a fundamental concept in robotics that represents the relationship between joint velocities and end-effector velocities in a robotic system. It serves as a crucial tool for analyzing motion, controlling robotic systems, and understanding how changes in joint parameters affect the position and orientation of the end effector. This matrix is vital for tasks such as velocity kinematics, dynamic modeling, workspace analysis, and gait planning for legged robots.
Jacobian Transpose: The Jacobian transpose is a mathematical construct that relates the velocities of the end effector of a robotic system to the joint velocities through the transpose of the Jacobian matrix. This concept is crucial in determining how changes in joint configurations influence the motion of the robot’s end effector, especially in velocity kinematics and static force analysis.
Joint Torques: Joint torques refer to the rotational forces that are applied at the joints of a robotic system, enabling movement and control of the robot's limbs. These torques are essential for the proper functioning of robots, as they directly relate to the forces required to achieve desired velocities and maintain static positions. Understanding joint torques allows for the effective design and operation of robotic systems by linking kinematic motion with force production.
Joint velocities: Joint velocities refer to the rates at which the angles or positions of the joints in a robotic system change over time. This concept is crucial for understanding how a robot moves and interacts with its environment, as it directly influences the robot's overall speed and maneuverability. Joint velocities are particularly important when analyzing motion control and ensuring that robotic arms and legs can execute tasks smoothly and accurately.
Linear Velocity Jacobian: The Linear Velocity Jacobian is a mathematical representation that relates the joint velocities of a robotic manipulator to the linear velocities of its end-effector. It captures how changes in joint positions influence the position and movement of the end-effector in Cartesian space. This concept is crucial for understanding motion control, trajectory planning, and how forces can be applied in robotic systems.
Manipulability Ellipsoid: The manipulability ellipsoid is a geometric representation that describes how easily a robotic manipulator can move in different directions at a given configuration. It highlights the robot's ability to exert force or achieve velocity in various directions, providing insights into its dexterity and workspace. The shape and size of the ellipsoid are influenced by the manipulator's configuration, joint limits, and physical characteristics, helping to identify optimal poses for tasks.
Manipulability measure: The manipulability measure is a quantitative metric used to evaluate the ease with which a robotic manipulator can achieve desired end-effector motions in relation to its configuration and workspace. This concept is essential for understanding how the geometry of a manipulator affects its performance, particularly in terms of velocity kinematics and static forces. It informs decisions about design and control strategies, enabling robots to execute tasks effectively by identifying optimal configurations that maximize motion capabilities while minimizing the risk of singularities.
Path Planning: Path planning is the process of determining a feasible and optimal trajectory for a robot to follow in order to move from a starting point to a target destination while avoiding obstacles. It involves using various algorithms and techniques to compute the best path that satisfies constraints such as distance, safety, and efficiency, making it crucial for effective robot navigation and operation.
Pseudo-inverse of Jacobian: The pseudo-inverse of the Jacobian is a mathematical tool used to find a solution to systems of linear equations, especially when these systems are underdetermined or overdetermined. In the context of velocity kinematics and static forces, it allows for the calculation of joint velocities from end-effector velocities in robotic systems, effectively linking the robot's configuration with its movement in space.
Static Force Analysis: Static force analysis is a method used to determine the forces acting on a mechanical system that is in equilibrium, meaning that the sum of forces and moments acting on it are zero. This analysis is crucial for understanding how structures and mechanisms will behave under various loads, ensuring their stability and safety in operation. By analyzing the forces, engineers can predict how a system will respond to static loads and make informed design decisions.
Task space dimensions: Task space dimensions refer to the parameters that define the workspace of a robotic system, encompassing the physical positions and orientations that a robot's end effector can achieve. Understanding task space dimensions is crucial for analyzing velocity kinematics and static forces as they help determine how effectively a robot can interact with its environment and perform various tasks, influencing design choices and control strategies.
Trajectory optimization: Trajectory optimization is the process of finding the best path or trajectory for a robotic system to follow in order to achieve a specific goal while minimizing costs such as time, energy, or deviation from constraints. This involves analyzing the motion dynamics and constraints of the system, which is crucial for effective control and performance.
Velocity Kinematics Equation: The velocity kinematics equation is a mathematical representation that relates the velocity of an object to its acceleration and the time over which this acceleration occurs. This equation helps to describe how the position of an object changes over time under constant acceleration, connecting concepts like displacement, initial velocity, final velocity, and time. Understanding this equation is crucial in analyzing the motion of objects and determining their trajectory and behavior in various mechanical systems.
Velocity Transmission Capabilities: Velocity transmission capabilities refer to the ability of a mechanical system to transfer input velocity to output velocity, while maintaining the effectiveness of movement throughout the system. This concept is essential in understanding how components like gears, belts, and linkages interact to control motion, ensuring that the desired speed and direction are achieved in robotic systems and mechanisms.
Virtual Work Principle: The virtual work principle states that for a system in equilibrium, the work done by the external forces during a virtual displacement is equal to the work done by the internal forces. This concept is crucial for analyzing mechanical systems, as it allows for the calculation of forces and displacements without needing to know the exact nature of the internal forces involved. By applying this principle, one can derive equations of motion and conditions for static equilibrium, which are foundational in understanding both velocity kinematics and static forces.
Workspace analysis: Workspace analysis is the process of determining the reachable and effective operational space of a robotic system, helping to identify the positions and orientations in which the robot can perform tasks. This analysis is crucial for understanding limitations and optimizing designs, as it relates directly to how effectively a robot can operate within its environment while considering factors like joint configurations and kinematic constraints.
Wrist Singularity: Wrist singularity refers to a condition in robotic arms where the wrist joint becomes aligned in a way that limits the arm's ability to control the end effector's orientation. This situation creates a scenario where the robot loses a degree of freedom, making it difficult to perform specific movements or achieve desired poses. Understanding wrist singularity is crucial for analyzing velocity kinematics and static forces, as it affects the robot's ability to execute tasks effectively without encountering control issues.