Notes in Week 4- Jacobian and Dynamics

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Status	Last Update	Fields
Published	11/26/2024	{{c1::Velocity kinematics}} establishes the relationship between the linear and angular velocities of the end-effector and the joint velocities.
Published	11/26/2024	The Jacobian matrix generalizes the notion of the {{c1::derivative}} to relate joint rates to end-effector velocities.
Published	11/26/2024	In velocity kinematics, the Jacobian matrix is derived from the {{c1::forward kinematic}} equations.
Published	11/26/2024	The {{c1::linear velocity}} of any point on a rigid body in rotation is given by \( \mathbf{v} = \mathbf{\omega} \times \mathbf{r} \).
Published	11/26/2024	The angular velocity of a rigid body rotating about a fixed axis is given by {{c1::\( \mathbf{\omega} = \dot{\theta} \mathbf{k} \),}} where \( \mathbf…
Published	11/26/2024	The objective in velocity kinematics is to relate the {{c1::linear}} and {{c2::angular}} velocities of the end-effector to joint velocities.
Published	11/26/2024	For an \( n \)-link manipulator, the {{c1::Jacobian}} \( \mathbf{J} \) is a matrix mapping joint velocities \( \dot{\mathbf{q}} \) to end-ef…
Published	11/26/2024	The angular velocity of the end-effector can be written as {{c2::\( \mathbf{\omega} = \mathbf{J}_\omega \dot{\mathbf{q} } \),}} with \( \mathbf{J…
Published	11/26/2024	The combined expression {{c1::\( \mathbf{\nu} = \mathbf{J} \dot{\mathbf{q} } \)}} involves both linear and angular velocities through the composite Ja…
Published	11/26/2024	For a rigid body in rotation, every point moves in a circle centered on the {{c1::axis of rotation}}.
Published	11/26/2024	In a manipulator, the linear Jacobian \( \mathbf{J}_v \) maps joint velocities to {{c1::end-effector linear velocities}}.
Published	11/26/2024	The angular Jacobian \( \mathbf{J}_\omega \) maps joint velocities to the {{c1::angular velocity}} of the end-effector.
Published	11/26/2024	For prismatic joints, the angular velocity \( \mathbf{\omega} \) is {{c1::zero}} since there is only translational movement.
Published	11/26/2024	In an \( m \)-DOF manipulator, \( \mathbf{J}_v \) and \( \mathbf{J}_\omega \) are {{c1::\( 3 \times m \)}} matrices.
Published	11/26/2024	The angular Jacobian \( \mathbf{J}_\omega \) for a prismatic joint is simply {{c1::zero}}.
Published	11/26/2024	For a simple RP manipulator, the coordinates of points \( O_1 \) and \( O_2 \) depend on \( \theta \) and \( d \), which represent {{c1::rotation}} an…
Published	11/26/2024	In robotic kinematics, the linear Jacobian component \( \mathbf{J}_v \) relates {{c1::joint velocities}} to the linear velocity of the {{c2::end effec…
Published	11/26/2024	The Jacobian matrix \( \mathbf{J} \) generally has dimensions \( 6 \times n \) for an \( n \)-joint manipulator, where the six rows correspond to {{c1…
Published	11/26/2024	In the Jacobian matrix, each column corresponds to the contribution of a single {{c1::joint}} to the total end-effector velocity.
Published	11/26/2024	The angular velocity part of the Jacobian \( \mathbf{J}_\omega \) maps joint velocities to the {{c1::angular velocity}} of the end-effector.
Published	11/26/2024	A Jacobian matrix that is full rank ensures that the manipulator can achieve velocities in {{c1::all directions}} at the end effector.
Published	11/26/2024	The Jacobian matrix helps in determining the range of motions available to the end effector, which is also known as the manipulator’s {{c1::workspace}…
Published	11/26/2024	If the Jacobian matrix is singular, certain directions of end-effector movement cannot be achieved due to a reduction in the manipulator’s {{c1::degre…
Published	11/26/2024	Singularities in robotic kinematics refer to configurations where the Jacobian loses rank, leading to {{c1::unachievable directions}} of end-effector …
Published	11/26/2024	The condition number of the Jacobian matrix provides insight into how close the manipulator is to a {{c1::singularity}}.
Published	11/26/2024	The Jacobian transpose method is often used in control to relate forces and torques at the joints to {{c1::forces and torques at the end effector}}.
Published	11/26/2024	In force control, the Jacobian transpose \( \mathbf{J}^T \) is used to map {{c1::end-effector forces}} to joint torques.
Published	11/26/2024	When the Jacobian is singular, certain end-effector velocities may require {{c1::infinite joint velocities}} to achieve, which is physically impractic…
Published	11/26/2024	At singularities, the manipulator’s ability to exert force in certain directions is {{c1::restricted}}, impacting its ability to manipulate objects.
Published	11/26/2024	The Jacobian can be divided into submatrices \( \mathbf{J}_v \) and \( \mathbf{J}_\omega \), where \( \mathbf{J}_v \) handles {{c1::linear velocity}} …
Published	11/26/2024	A well-conditioned Jacobian indicates that the manipulator has effective movement and force control, whereas a poorly conditioned Jacobian suggests {{…
Published	11/26/2024	Manipulators reach kinematic singularities when the Jacobian's rank drops, limiting the range of achievable {{c1::velocities and forces}} at the end e…
Published	11/26/2024	Jacobian-based control methods are valuable in robotic systems as they enable accurate and controlled {{c1::motion planning}} and force application.
Published	11/26/2024	The Jacobian matrix is central to the study of kinematics, allowing engineers to model how joint motion translates into {{c1::end-effector motion}}.
Published	11/26/2024	In velocity kinematics, the Jacobian inverse can be used to calculate {{c1::joint velocities}} needed to achieve a specified end-effector velocity.
Published	11/26/2024	The Jacobian pseudo-inverse is often used in redundant manipulators to find a {{c1::least-squares solution}} for joint velocities that achieve a desir…
Published	11/26/2024	The Euler-Lagrange equation is used to derive equations of motion and is given by \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_k} \right)…
Published	11/26/2024	In dynamics, the Lagrangian \( L \) is defined as \( L = K - P \), where \( K \) is the {{c1::kinetic energy}} and \( P \) is the {{c2::potential ener…
Published	11/26/2024	For a system of \( n \) links, the Lagrangian approach requires calculating kinetic and potential energy in terms of {{c1::generalized coordinates}}.
Published	11/26/2024	The term \( \frac{\partial L}{\partial \dot{q}_k} \) represents the derivative of the Lagrangian with respect to the {{c1::generalized velocity}} \( \…
Published	11/26/2024	The kinetic energy \( K \) for a rigid body is given by \( K = \frac{1}{2} m \mathbf{v}^T \mathbf{v} + \frac{1}{2} \omega^T I \omega \), where \( I \)…
Published	11/26/2024	The Lagrangian method combines both energy concepts, making it more adaptable for complex, {{c1::multi-body}} systems.
Published	11/26/2024	The gravitational force in dynamics is related to potential energy as \( m g = \frac{\partial P}{\partial y} \), where \( P \) is the {{c1::gravitatio…
Published	11/26/2024	For a rigid body, the translational kinetic energy considers the body’s entire mass concentrated at the {{c1::center of mass}}.
Published	11/26/2024	The equation of motion for a particle in one direction, given by Newton’s second law, is \( m \ddot{y} = f - mg \), where \( f \) is the {{c1::externa…
Published	11/26/2024	The inertia tensor \( \mathbb{I} \) in dynamics represents the rotational resistance of a body and is calculated using a {{c1::3x3 symmetric matrix}}.
Published	11/26/2024	The inertia matrix is expressed in the body frame and remains constant, independent of the body’s {{c1::motion}}.
Published	11/26/2024	For an \( n \)-link manipulator, the total potential energy \( P \) is given by the sum \( P = \sum_{i=1}^n m_i g^T r_{ci} \), where \( r_{ci} \) is t…
Published	11/26/2024	The kinetic energy of a rigid body is the sum of the {{c1::translational kinetic energy}} and {{c2::rotational kinetic energy}}.
Published	11/26/2024	In terms of potential energy, the gravitational component for a link \( i \) is \( P_i = m_i g^T r_{ci} \), which considers the body’s mass at its {{c…
Published	11/26/2024	The inertia tensor \( \mathbb{I} \) undergoes a similarity transformation in a rotated frame, given by \( \mathbb{I} = RIR^T \), where \( R \) is the …
Published	11/26/2024	The general form of kinetic energy for an \( n \)-link manipulator is \( K = \frac{1}{2} \dot{q}^T D(q) \dot{q} \), where \( D(q) \) is the {{c1::iner…
Published	11/26/2024	The inertia matrix \( D(q) \) is positive definite, ensuring stability in the robot’s {{c1::dynamic equations}}.
Published	11/26/2024	The generalized force vector \( \tau \) in the Euler-Lagrange equation accounts for both {{c1::applied forces}} and {{c2::gravitational forces}}.
Published	11/26/2024	In multi-link systems, the total kinetic energy \( K \) combines the kinetic energy contributions of both linear and {{c1::angular motions}} of each l…
Published	11/26/2024	The Euler-Lagrange equation simplifies to \( D(q) \ddot{q} + C(q, \dot{q}) \dot{q} + G(q) = f \), where \( C(q, \dot{q}) \) includes {{c1::Coriolis an…
Published	11/26/2024	The Coriolis matrix \( C(q, \dot{q}) \) appears in the dynamic equations of motion to account for effects from {{c1::rotating reference frames}}.
Published	11/26/2024	The term \( G(q) = \frac{\partial P}{\partial q} \) in the equations of motion represents the forces due to {{c1::potential energy}}, often gravity.
Published	11/26/2024	The equations of motion are derived by calculating the partial derivatives of the Lagrangian with respect to each {{c1::generalized coordinate}}.
Published	11/26/2024	In a two-link planar manipulator, the inertia matrix \( D(q) \) contains terms that depend on link lengths, masses, and the {{c1::angles between links…
Published	11/26/2024	The total potential energy of a multi-link robot is the sum of the potential energies of each link, concentrated at their {{c1::centers of mass}}.
Published	11/26/2024	In planar manipulators, the Coriolis and centrifugal forces result in a non-zero \( C(q, \dot{q}) \) matrix, affecting {{c1::joint torques}} during mo…
Published	11/26/2024	The term \( D(q) \dot{q} \) represents the inertia forces and appears as a {{c1::mass matrix}} multiplied by the joint accelerations in the equation o…
Published	11/26/2024	Centrifugal forces in the motion equations arise from the joint’s rotational motion and depend on the {{c1::velocity}} of each link.
Published	11/26/2024	The inertia matrix \( D(q) \) in robotic dynamics is symmetric and positive-definite, representing the system's {{c1::resistance to acceleration}}.
Published	11/26/2024	Gravity compensation in robotic manipulators reduces the load on actuators by using mechanisms like {{c1::counterweights}} and {{c2::springs}}.
Published	11/26/2024	The five-bar linkage is used in robotics for applications requiring a compact design with high {{c1::mechanical stability}} and precise movements.
Published	11/26/2024	Gravity compensation aims to balance the robot’s weight throughout its range of motion, often reducing {{c1::actuator power requirements}}.
Published	11/26/2024	Gravity compensation allows robotic arms to {{c2::hold positions}} without using continuous actuator power, which is essential for {{c1::energy effici…
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