Robotics PC 5: Velocity and Jacobian

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What is the primary method used to solve direct velocity problems in robotic manipulators?

Vector propagation

In the context of velocity propagation, what is the significance of the frame in which the velocity is represented?

It affects the calculation of velocities and angular velocities

What is the purpose of the Jacobian in robotic manipulators?

To map joint velocity inputs to the point velocity of the end-effector tip

In which two frames can the Jacobian be represented for the 3-link manipulators?

𝔽0 and 𝔽3 Signup and view all the answers

What happens to the rank of a square Jacobian matrix under singularity configuration?

It becomes less than full rank Signup and view all the answers

Do singularities in the force domain exist at the same configuration as singularities in the position domain?

No Signup and view all the answers

What is the physical meaning of a singularity in the force domain?

The inability to apply forces in certain directions Signup and view all the answers

What is the Jacobian of the end-effector in the tool frame, i.e. 𝐽4?

𝐽4 = [∂x4/∂θ1 ∂x4/∂θ2 ∂x4/∂θ3; ∂y4/∂θ1 ∂y4/∂θ2 ∂y4/∂θ3; 0 0 0] Signup and view all the answers

What is the importance of considering the angular velocity of the end-effector in certain applications?

It is crucial for precise control and movement Signup and view all the answers

What is the Jacobian of the end-effector in the base frame, i.e. 𝐽0?

𝐽0 = [∂x0/∂θ1 ∂x0/∂θ2 ∂x0/∂θ3; ∂y0/∂θ1 ∂y0/∂θ2 ∂y0/∂θ3; 0 0 0] Signup and view all the answers

How can the joint torques required to maintain a static force vector 𝐹 be represented?

τ = 𝐽0^T 𝐹 Signup and view all the answers

What is the transformation matrix ¹𝑇 between the base and tool frame?

¹𝑇 = [𝑐1 -𝑠1 0 0 1 0; 0 𝑠1 𝑐1 0 1 0; 0 0 1 0 1 0; 0 0 0 1 0 1] Signup and view all the answers

What is the transformation matrix ²𝑇 between the base and tool frame?

²𝑇 = [𝑐1 𝑠1 0 0 1 0; -𝑠1 𝑐1 0 0 1 0; 0 0 1 0 1 0; 0 0 0 1 0 1] Signup and view all the answers

What is the velocity (linear and angular) of the end-effector in the base frame?

⁰𝑣₅ = 𝐽₀ ⋅ [θ̇₁ θ̇₂ θ̇₃]ᵀ, ⁰𝜔₅ = [₀𝜔x₅ ₀𝜔y₅ ₀𝜔z₅]ᵀ Signup and view all the answers

What is the time differentiation method used for in robotics?

Finding the velocity and acceleration of the end-effector Signup and view all the answers

What is the importance of the transformation matrices in robotics?

To transform coordinates between different frames Signup and view all the answers

What is the orientation of the working platform in the personnel lifter mechanism?

Always oriented in the direction of 𝑧̂0 Signup and view all the answers

What type of joint is joint 3 in the personnel lifter mechanism?

Prismatic Signup and view all the answers

How does a load in any direction on the platform translate to the fourth joint?

It translates to a point force in the same direction coincident at the fourth joint. Signup and view all the answers

What are the torques and prismatic joint force needed to support any force 0𝐹 applied to the platform?

𝜏1, 𝜏2, and 𝐹3 Signup and view all the answers

What is the significance of the singularities of this system?

They physically represent the configurations where the mechanism loses its ability to support any force. Signup and view all the answers

What is the purpose of the velocity propagation method in this problem?

To find the velocity (linear and angular) of the end-effector in the tool frame. Signup and view all the answers

What is the expression for the angular velocity of a prismatic joint using the velocity propagation method?

𝑖+1 𝜔𝑖+1 = 𝑖 𝑅 𝜔𝑖 Signup and view all the answers

What is the expression for the linear velocity of a prismatic joint using the velocity propagation method?

𝑖+1 𝑣𝑖+1 = 𝑖+1 𝑖𝑅 ( 𝑖𝑣𝑖 + 𝑖𝜔𝑖 × 𝑖𝑃𝑖+1 ) + 𝑑̂𝑖+1 𝑖+1 𝑍̂𝑖+1 Signup and view all the answers

What are the two components that the Jacobian of the end-effector in the base frame (𝐽0) includes?

Linear and angular velocity Signup and view all the answers

What is the physical interpretation of the singularities of the robot?

Positions where the robot loses one or more degrees of freedom. Signup and view all the answers

What is the purpose of finding the velocity (linear and angular) of the end-effector in the tool frame?

To determine the movement of the end-effector in the tool frame. Signup and view all the answers

What are the transformation matrices used for in the robot's movement?

To describe the movement of the robot's joints and end-effector. Signup and view all the answers

What is the significance of the tool frame in the robot's movement?

It is a reference frame attached to the end-effector. Signup and view all the answers

What is the purpose of finding the Jacobian of the end-effector in the tool frame (𝐽5)?

To relate the joint velocities to the end-effector velocities in the tool frame. Signup and view all the answers

What is the symbolic equation which represents the workspace boundary of the robot?

Depends on the specific robot's geometry and configuration. Signup and view all the answers

What are 𝑐12−4 and 𝑠12−4 in the transformation matrices?

Trigonometric functions of the joint angles (cos(𝜃1 + 𝜃2 − 𝜃4) and sin(𝜃1 + 𝜃2 − 𝜃4) respectively). Signup and view all the answers

Using the time derivative of the point position, derive the point velocity and angular velocity, represented in 𝐹0 (frame 0) of the end-effector of the three-link manipulator shown in Figure 1.

The point velocity and angular velocity can be derived by taking the time derivative of the point position in frame 0, which involves differentiating the position vector with respect to time. Signup and view all the answers

Using the velocity propagation method, derive the point velocity and angular velocity, represented in 𝔽4 (frame 4) of the end-effector of the three-link manipulator shown in Figure 2.

The point velocity and angular velocity can be derived by propagating the velocity from the base frame to the end-effector frame using the velocity propagation method. Signup and view all the answers

Derive the Jacobian for the manipulator mapping the joint velocity inputs to the point velocity of the end-effector tips for the 3 link manipulators from Figure 1 and 2.

The Jacobian can be derived by differentiating the forward kinematics equation with respect to time, resulting in a matrix that maps joint velocities to end-effector velocities. Signup and view all the answers

What happens to the rank of a square Jacobian matrix under singularity configuration?

The rank of a square Jacobian matrix reduces under singularity configuration, resulting in a loss of manipulability. Signup and view all the answers

Explain the physical meaning of a singularity in the force domain.

A singularity in the force domain occurs when the manipulator is unable to generate forces in certain directions, resulting in a loss of force manipulability. Signup and view all the answers

How can the joint torques required to maintain a static force vector 𝐹 be represented?

The joint torques required to maintain a static force vector 𝐹 can be represented using the Jacobian transpose and the force vector. Signup and view all the answers

What is the significance of the transformation matrices in robotics?

The transformation matrices are used to describe the relationships between different frames of reference in robotic manipulators, enabling the transformation of velocities and forces between frames. Signup and view all the answers

What is the physical interpretation of the singularities of the robot?

The physical interpretation of the singularities of the robot is a loss of manipulability, resulting in a reduced ability to move freely or apply forces in certain directions. Signup and view all the answers

Derive the expression for the linear velocity of the end-effector in the tool frame using the velocity propagation method for the robot shown in Figure 3.

4𝑣4 = ⁴𝑅₃(³𝑣₃ + ³𝜔₃ × ³𝑃₄) + 𝑑̂₄ ⁴𝑍̂₄ Signup and view all the answers

Find the angular velocity of the end-effector in the tool frame using the velocity propagation method for the robot shown in Figure 3.

4𝜔4 = ⁴𝑅₃(³𝜔₃) Signup and view all the answers

Derive the expression for the transformation matrix ⁴𝑇 between the base and tool frame for the robot shown in Figure 3.

⁴𝑇 = ⁴𝑅₀(⁰𝑣₀ + ⁰𝜔₀ × ⁰𝑃₄) + 𝑑̂₄ ⁴𝑍̂₄ Signup and view all the answers

Find the Jacobian of the end-effector in the tool frame (𝐽₄) for the robot shown in Figure 3.

𝐽₄ = [⁴𝑅₃(³𝑣₃ + ³𝜔₃ × ³𝑃₄) + 𝑑̂₄ ⁴𝑍̂₄, ⁴𝑅₃(³𝜔₃)] Signup and view all the answers

Find the Jacobian of the end-effector in the base frame (𝐽₀) for the robot shown in Figure 3.

𝐽₀ = [⁰𝑅₄(⁴𝑣₄ + ⁴𝜔₄ × ⁴𝑃₀) + 𝑑̂₄ ⁰𝑍̂₄, ⁰𝑅₄(⁴𝜔₄)] Signup and view all the answers

Derive the expression for the torques and prismatic joint force needed to support any force 0𝐹 applied to the platform for the robot shown in Figure 3.

𝜏₁ = 𝑓𝑥, 𝜏₂ = 𝑓𝑦, 𝐹₃ = 𝑓𝑧 Signup and view all the answers

What are the singularities of this system and what do they physically mean?

The singularities of this system occur when the Jacobian matrix becomes singular, and physically mean that the robot is unable to move in certain directions. Signup and view all the answers

Find the velocity (linear and angular) of the end-effector in the base frame for the robot shown in Figure 4.

⁰𝑣₄ = ⁰𝑅₄(⁴𝑣₄ + ⁴𝜔₄ × ⁴𝑃₀) + 𝑑̂₄ ⁰𝑍̂₄, ⁰𝜔₄ = ⁰𝑅₄(⁴𝜔₄) Signup and view all the answers

What is the Jacobian of the end-effector in the tool frame, i.e. 𝐽4, in terms of the linear velocity terms?

𝐽4 = ∂𝑥4/∂𝑞 = [−𝐿1𝑠1 𝐿1𝑐1 0; 𝐿3𝑠3 𝐿3𝑐3 0; 0 0 1] Signup and view all the answers

What is the Jacobian of the end-effector in the base frame, i.e. 𝐽0, in terms of the linear velocity terms?

𝐽0 = 𝐽4 × ⁴𝑇 = [−𝐿1𝑠1 𝐿1𝑐1 0; 𝐿3𝑠3 𝐿3𝑐3 0; 0 0 1] × ⁴𝑇 Signup and view all the answers

How can the joint torques required to maintain a static force vector 𝐹 be represented as a matrix equation?

τ = 𝐽0^T × 𝐹, where τ is the joint torque vector and 𝐽0^T is the transpose of the Jacobian matrix 𝐽0 Signup and view all the answers

What is the velocity (linear and angular) of the end-effector in the base frame, i.e. ⁰𝑣5 & ⁰𝜔5, using the time differentiation method?

⁰𝑣5 = 𝐽0 × 𝑞̇, ⁰𝜔5 = 𝐽0 × 𝑞̇, where 𝑞̇ is the joint velocity vector Signup and view all the answers

What is the transformation matrix ⁴𝑇 between the base and tool frame?

⁴𝑇 = [𝑐1 −𝑠1 0 0; 𝑠1 𝑐1 0 0; 0 0 1 0] × [𝑐3 −𝑠3 0 0; 𝑠3 𝑐3 0 0; 0 0 1 0] Signup and view all the answers

What is the expression for the linear velocity of the end-effector in the tool frame, i.e. ⁴𝑣5, using the time differentiation method?

⁴𝑣5 = 𝐽4 × 𝑞̇, where 𝐽4 is the Jacobian matrix in the tool frame and 𝑞̇ is the joint velocity vector Signup and view all the answers

How can the singularities of the robot be physically interpreted?

Singularities occur when the robot's joints align in a way that the end-effector's velocity is undefined or cannot be controlled Signup and view all the answers

What is the significance of the Jacobian matrix in robotic manipulators?

The Jacobian matrix represents the linear and angular velocity of the end-effector in terms of the joint velocities, and is used to solve direct and inverse kinematics problems Signup and view all the answers

Derive the velocity (linear and angular) of the end-effector in the tool frame, i.e. 5𝑣5 and 5𝜔5, using the transformation matrices provided.

5𝑣5 = [𝐿1 𝑠1 + 𝐿2 𝑠12, 𝐿1 𝑐1 + 𝐿2 𝑐12, 0]; 5𝜔5 = [0, 0, 1] Signup and view all the answers

Find the Jacobian of the end-effector in the base frame, i.e. 𝐽0, including both terms for linear and angular velocity.

𝐽0 = [∂𝑥0/∂𝜃1, ∂𝑥0/∂𝜃2, ∂𝑥0/∂𝜃3, ∂𝑥0/∂𝜃4; ∂𝜔0/∂𝜃1, ∂𝜔0/∂𝜃2, ∂𝜔0/∂𝜃3, ∂𝜔0/∂𝜃4] Signup and view all the answers

Find the Jacobian of the end-effector in the tool frame, i.e. 𝐽5, including both terms for linear and angular velocity.

𝐽5 = [∂𝑥5/∂𝜃1, ∂𝑥5/∂𝜃2, ∂𝑥5/∂𝜃3, ∂𝑥5/∂𝜃4; ∂𝜔5/∂𝜃1, ∂𝜔5/∂𝜃2, ∂𝜔5/∂𝜃3, ∂𝜔5/∂𝜃4] Signup and view all the answers

Find the singularities of the robot, giving a physical interpretation of them.

The singularities occur when the Jacobian matrix becomes singular, i.e. when the determinant of the Jacobian is zero. Physically, this means that the robot is unable to move in certain directions. Signup and view all the answers

Find the symbolic equation which represents the workspace boundary of the robot.

The workspace boundary can be represented by the equation: 𝑥² + 𝑦² + 𝑧² = 𝐿1² + 𝐿2² + 𝐿4² Signup and view all the answers

Derive the expression for the linear velocity of the end-effector in the tool frame using the transformation matrices.

5𝑣5 = 𝐽5 𝜃̇, where 𝐽5 is the Jacobian of the end-effector in the tool frame and 𝜃̇ is the joint velocity Signup and view all the answers

Derive the expression for the angular velocity of the end-effector in the tool frame using the transformation matrices.

5𝜔5 = 𝐽5 𝜃̇, where 𝐽5 is the Jacobian of the end-effector in the tool frame and 𝜃̇ is the joint velocity Signup and view all the answers

Explain the importance of considering the angular velocity of the end-effector in certain applications.

The angular velocity of the end-effector is important in applications where the orientation of the end-effector is critical, such as in assembly or welding tasks. Signup and view all the answers

Study Notes

PC 5: Velocity, Jacobian, and Statics

Problem 1: Deriving Point Velocity and Angular Velocity

Derive the point velocity and angular velocity of the end-effector of a 3-link manipulator shown in Figure 1 using the time derivative of the point position.
The manipulator is represented in frame 0 (𝐹0).

Problem 2: Deriving Point Velocity and Angular Velocity using Velocity Propagation

Derive the point velocity and angular velocity of the end-effector of a 3-link manipulator shown in Figure 2 using the velocity propagation method.
Assume the length of the last link is L3.
The manipulator is represented in frame 4 (𝔽4), which is located at the tip of the end-effector, with the same orientation as 𝔽3.

Problem 3: Deriving the Jacobian

Derive the Jacobian for the manipulators mapping the joint velocity inputs to the point velocity of the end-effector tips for both the 3-link manipulators from Figure 1 and 2.
Represent the Jacobian in:
- 𝔽0 attached to the base.
- 𝔽3 attached to the 3rd link.
The angular velocity of the end-effector is not required.

Problem 4: Singularity of a Square Jacobian Matrix

Describe what happens to the rank of a square Jacobian matrix under singularity configuration.
Explain if singularities in the force domain exist at the same configuration as singularities in the position domain.
Explain the physical meaning of a singularity in the force domain.

Problem 5: Simplified Model of a Personnel Lifter Mechanism

Analyze a simplified model of a personnel lifter mechanism with a working platform as the end-effector, which is always oriented in the direction of 𝑧̂0.
Joints 1, 2, and 4 are revolute, with the first two coincident and the third joint prismatic, offset from joint 2 by distance L with variable extension d.
Find equations for the torques, 𝜏1, 𝜏2, and prismatic joint force 𝐹3, such that the mechanism can support any force 0𝐹 = [𝑓𝑥 𝑓𝑦 𝑓𝑧]𝑇, applied to the platform.
Identify the singularities of this system and their physical meaning.

Problem 6: Velocity and Jacobian of a 4-Dof RRPR Mechanism

Using the propagation method, find the velocity (linear and angular) of the end-effector in the tool frame, i.e., find 4𝑣4 and 4𝜔4.
Using the results, find the velocity (linear and angular) of the end-effector in the base frame, i.e., find 0𝑣4 and 0𝜔4.
Find the Jacobian of the end-effector in the tool frame, i.e., 𝐽4 (include only linear velocity terms).
Find the Jacobian of the end-effector in the base frame, i.e., 𝐽0 (include only linear velocity terms).
Find the joint torques required to maintain a static force vector 𝐹 = [𝑓𝑥 𝑓𝑦 0]𝑇, represented as a matrix equation.

Problem 7: Velocity and Jacobian of an Orthopaedic Robot

Using the time differentiation method, find the velocity (linear and angular) of the end-effector in the base frame, i.e., find 0𝑣5 and 0𝜔5.
Using the results, find the velocity (linear and angular) of the end-effector in the tool frame, i.e., find 5𝑣5 and 5𝜔5.
Find the Jacobian of the end-effector in the base frame, i.e., 𝐽0 (include both terms for linear and angular velocity).
Find the Jacobian of the end-effector in the tool frame, i.e., 𝐽5 (include both terms for linear and angular velocity).
Find the singularities (if any) of the robot, giving a physical interpretation of them.
Find the symbolic equation that represents the workspace boundary of the robot.