Robotics PC 5: Velocity and Jacobian

Vector propagation

It affects the calculation of velocities and angular velocities

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

<p>𝔽0 and 𝔽3</p> Signup and view all the answers

<p>It becomes less than full rank</p> Signup and view all the answers

<p>No</p> Signup and view all the answers

<p>The inability to apply forces in certain directions</p> Signup and view all the answers

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

<p>It is crucial for precise control and movement</p> Signup and view all the answers

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

<p>τ = 𝐽0^T 𝐹</p> Signup and view all the answers

<p>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]</p> Signup and view all the answers

<p>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]</p> Signup and view all the answers

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

<p>Finding the velocity and acceleration of the end-effector</p> Signup and view all the answers

<p>To transform coordinates between different frames</p> Signup and view all the answers

<p>Always oriented in the direction of 𝑧̂0</p> Signup and view all the answers

<p>Prismatic</p> Signup and view all the answers

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

<p>𝜏1, 𝜏2, and 𝐹3</p> Signup and view all the answers

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

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

<p>𝑖+1 𝜔𝑖+1 = 𝑖 𝑅 𝜔𝑖</p> Signup and view all the answers

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

<p>Linear and angular velocity</p> Signup and view all the answers

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

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

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

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

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

<p>Depends on the specific robot's geometry and configuration.</p> Signup and view all the answers

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

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

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

<p>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.</p> Signup and view all the answers

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

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

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

<p>4𝜔4 = ⁴𝑅₃(³𝜔₃)</p> Signup and view all the answers

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

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

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

<p>𝜏₁ = 𝑓𝑥, 𝜏₂ = 𝑓𝑦, 𝐹₃ = 𝑓𝑧</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

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

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

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

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

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

<p>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]</p> Signup and view all the answers

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

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

<p>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</p> Signup and view all the answers

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

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

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

<p>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.</p> Signup and view all the answers

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

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

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

<p>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.</p> Signup and view all the answers