systems.builtin.deterministic.continuous.Manipulator2Link

systems.builtin.deterministic.continuous.Manipulator2Link(
    m1=1.0,
    m2=1.0,
    l1=1.0,
    l2=1.0,
    lc1=0.5,
    lc2=0.5,
    I1=0.1,
    I2=0.1,
    gravity=9.81,
    friction1=0.1,
    friction2=0.1,
)

Two-link planar robotic manipulator - second-order formulation.

Physical System:

A planar robot arm with two revolute joints, each driven by a motor that applies torque.

The system consists of: - Two rigid links connected in series - Two revolute (rotational) joints at base and elbow - Actuators (motors) at each joint providing control torques - Gravity acting downward - Joint friction opposing motion

Configuration:

• Link 1: Attached to fixed base, length l₁, mass m₁
• Link 2: Attached to end of link 1, length l₂, mass m₂
• Joint 1 (base): Angle q₁ from horizontal
• Joint 2 (elbow): Angle q₂ relative to link 1

State Space:

State: x = [q₁, q₂, q̇₁, q̇₂] Joint angles (configuration): - q₁: Base joint angle [rad] * q₁ = 0: link 1 horizontal to the right * q₁ = π/2: link 1 pointing up - q₂: Elbow joint angle [rad] * q₂ = 0: links aligned (straight arm) * q₂ = π: fully bent (folded back)

Joint velocities:
- q̇₁: Base angular velocity [rad/s]
- q̇₂: Elbow angular velocity [rad/s]

Control: u = [τ₁, τ₂] - τ₁: Torque applied at base joint [N·m] - τ₂: Torque applied at elbow joint [N·m]

Output: y = [q₁, q₂] - Measures joint angles (typical for robots with encoders) - Does not directly measure end-effector position

Dynamics:

The manipulator dynamics follow the standard robot equation: M(q)q̈ + C(q,q̇)q̇ + G(q) + F(q̇) = τ

where: - M(q): Configuration-dependent inertia matrix (2×2) - C(q,q̇): Coriolis and centrifugal terms - G(q): Gravity terms - F(q̇): Friction terms - τ: Applied joint torques (control)

Inertia Matrix M(q): The inertia matrix captures how joint accelerations relate to torques. It depends on configuration due to changing mass distribution:

M₁₁ = m₁·lc₁² + m₂·(l₁² + lc₂² + 2·l₁·lc₂·cos(q₂)) + I₁ + I₂
M₁₂ = m₂·(lc₂² + l₁·lc₂·cos(q₂)) + I₂
M₂₁ = M₁₂  (symmetric)
M₂₂ = m₂·lc₂² + I₂

Key features: - M is symmetric and positive definite - Diagonal terms: self-inertia of each joint - Off-diagonal: coupling between joints - M₁₁ maximized when arm is extended (q₂ = 0)

Coriolis and Centrifugal Terms C(q,q̇): These arise from coordinate system rotation and create coupling:

h = -m₂·l₁·lc₂·sin(q₂)
C₁ = h·(2·q̇₁·q̇₂ + q̇₂²)
C₂ = -h·q̇₁²

Physical interpretation: - When joint 2 moves, it creates forces on joint 1 (and vice versa) - Centrifugal: pushing outward when rotating - Coriolis: deflection perpendicular to motion

Gravity Terms G(q): Gravitational torques trying to pull arm downward:

G₁ = (m₁·lc₁ + m₂·l₁)·g·cos(q₁) + m₂·lc₂·g·cos(q₁ + q₂)
G₂ = m₂·lc₂·g·cos(q₁ + q₂)

Key features: - Maximum when arm horizontal (cos = 1) - Zero when arm vertical (cos = 0) - Both links contribute to joint 1 torque - Only link 2 affects joint 2 torque

Friction F(q̇): Simple viscous friction model:

F₁ = b₁·q̇₁
F₂ = b₂·q̇₂

Solving for Accelerations: The forward dynamics gives: q̈ = M⁻¹(τ - C - G - F)

Parameters:

m1 : float, default=1.0 Mass of link 1 [kg]. Affects inertia and gravity torques. m2 : float, default=1.0 Mass of link 2 [kg]. Lighter distal link → faster motion. l1 : float, default=1.0 Length of link 1 [m]. Distance from base to elbow. l2 : float, default=1.0 Length of link 2 [m]. Distance from elbow to end-effector. lc1 : float, default=0.5 Distance from base joint to center of mass of link 1 [m]. Typically lc₁ = l₁/2 for uniform link. lc2 : float, default=0.5 Distance from elbow joint to center of mass of link 2 [m]. Typically lc₂ = l₂/2 for uniform link. I1 : float, default=0.1 Moment of inertia of link 1 about its center of mass [kg·m²]. For uniform rod: I = (1/12)·m·l² I2 : float, default=0.1 Moment of inertia of link 2 about its center of mass [kg·m²]. gravity : float, default=9.81 Gravitational acceleration [m/s²]. friction1 : float, default=0.1 Viscous friction coefficient at joint 1 [N·m·s/rad]. friction2 : float, default=0.1 Viscous friction coefficient at joint 2 [N·m·s/rad].

Equilibria:

Hanging down (stable): q_eq = [π, 0] (link 1 down, link 2 aligned) q̇_eq = [0, 0] (at rest) τ_eq = [0, 0] (gravity balances)

Horizontal (unstable without control): q_eq = [0, 0] (both links horizontal) Requires active control due to gravity

Upright (highly unstable): q_eq = [π/2, 0] (both links pointing up) Requires fast, precise control

Forward Kinematics:

End-effector position in Cartesian space: x_ee = l₁·cos(q₁) + l₂·cos(q₁ + q₂) y_ee = l₁·sin(q₁) + l₂·sin(q₁ + q₂)

Workspace: - Circle of radius l₁ + l₂ (maximum reach) - Inner circle of radius |l₁ - l₂| (unreachable) - Singularities when arm fully extended or folded

See Also:

CartPole : Similar dynamics but with prismatic (sliding) joint SymbolicPendulum2ndOrder : Single-link version PVTOL : Flying robot with similar multi-body coupling