4-DoF Ground-Aligned Planar Model

• The paper presents a 4-DoF parallel manipulator with two translational and two rotational degrees using a 3-UPS plus RPU architecture.
• It provides closed-form inverse and forward kinematics with detailed loop-closure equations and Jacobian analysis for singularity detection.
• The model is designed for lower limb rehabilitation, enabling precise control of force and torque in sagittal-plane movements.

A 4-DoF ground-aligned planar model refers to a parallel manipulator with two translational degrees of freedom and two rotational degrees of freedom in the vertical sagittal plane. This architecture, specifically realized as a 3-UPS (Universal-Prismatic-Spherical) plus RPU (Revolute-Prismatic-Universal) configuration, delivers closed-form solutions for both inverse and forward kinematics, and is notable for its utility in lower limb rehabilitation. The manipulator exploits planar motion, with explicitly defined coordinate frames and comprehensive symbolic kinematic parameters, supporting control and singularity analysis grounded in established conventions (Valles et al., 2024).

1. Kinematic Architecture and Coordinate Frames

The model employs two reference frames:

• The fixed (ground) frame $\{F\}$, with origin $O_F$; axes $X_F$ (horizontal), $Z_F$ (vertical), and $Y_F$ (out of plane).
• The moving-platform frame $\{m\}$, with origin $O_m$ at the center of the mobile platform; $X_m \parallel X_F$, $Z_m \parallel Z_F$, $Y_m \parallel Y_F$.

The end-effector pose is parameterized by four coordinates: $(x, z, \theta, \psi)$. Here, $x \equiv X_m$ and $z \equiv Z_m$ are the translations in the sagittal plane, while $\theta$ and $\psi$ denote rotations about $Y_F$ (pitch) and the platform normal (yaw), respectively.

Actuation is provided by three prismatic joints in the external legs ($q_{13}$, $q_{23}$, $q_{33}$ for legs $i=1,2,3$) and one prismatic joint in the central RPU leg ($q_{4,2}$). All other joints are passive and not explicitly used by the control law.

2. Symbolic Link Parameters (Paul’s D-H Convention)

The manipulator comprises four limbs attached via geometric constants:

Leg	Architecture	D-H Joint Sequence	Actuated Joints	Passive Joints
i=1-3	UPS	U-P-S (6 joints, see details below)	$q_{i3}$	All others
4	RPU	R-P-U (4 joints, see details below)	$q_{4,2}$	All others

Key geometric parameters:

• $r$: radius of attachment circle on the fixed base for external legs
• $r_m$: radius of attachment circle on the mobile platform
• $\beta_i = (i-1)\cdot120^{\circ}$: angular position of leg $i$

Paul’s Denavit-Hartenberg convention is used to define each joint’s $\alpha$, $a$, $d$, and $\theta$ values, with explicit distinction between the actuated prismatic joints and the passive revolute/spherical ones.

3. Closed-Form Inverse Kinematics

Given the desired pose $(x, z, \theta, \psi)$, the corresponding active joint variables are computed as follows.

• Central Leg (RPU):
  • $x = -\sin q_{4,1} \, q_{4,2}$
  • $z = \cos q_{4,1} \, q_{4,2}$
  • $q_{4,1} = \operatorname{atan2}(-x,\, z)$
  • $q_{4,2} = \sqrt{x^2 + z^2}$
• External Legs (UPS, $i=1,2,3$):
  • $A_i = [r \cos\alpha_i,\, 0,\, r \sin\alpha_i]^T$ (fixed-base joint in $\{F\}$)
  • $$B_i = [x, 0, z]^T + R_y(\theta)R_z(\psi)[r_m \cos\alpha_i,\, 0,\, r_m \sin\alpha_i]^T$$ (mobile-platform joint in $\{F\}$)
  • Leg vector $p_i = B_i - A_i$
  • Prismatic length:

  $$q_{i3} = \|p_i\| = \sqrt{[x + r_m \cos\psi_i \cos\theta - r \cos\alpha_i]^2 + [z + r_m \sin\psi_i \sin\theta - r \sin\alpha_i]^2}$$

  with $\psi_i = \psi + \alpha_i$ - Passive joint angles (for completeness, not actuated): - $q_{i1} = \operatorname{atan2}(p_{i,z},\, p_{i,x})$ - $$q_{i2} = \operatorname{atan2}(p_{i,x}\sin q_{i1} + p_{i,z}\cos q_{i1},\, p_{i,z}\sin q_{i1} - p_{i,x}\cos q_{i1})$$

4. Forward Kinematics via Loop-Closure Equations

The forward-displacement problem seeks $(x, z, \theta, \psi)$ for given actuator settings $\{q_{13}, q_{23}, q_{33}, q_{4,2}\}$. Seven passive joints are algebraically eliminated to yield four loop-closure equations:

For each external leg $i=1,2,3$:

$$F_i(x, z, \theta, \psi; q_{i3}) \equiv \| R_y(\theta) R_z(\psi) [r_m \cos\alpha_i, 0, r_m \sin\alpha_i]^T + [x, 0, z]^T - [r \cos\alpha_i, 0, r \sin\alpha_i]^T \| - q_{i3} = 0$$

Central leg (RPU):

$$F_4(x, z; q_{4,2}) \equiv \sqrt{x^2 + z^2} - q_{4,2} = 0$$

Numerical solution (e.g., Newton-Raphson) is effective, since all passive joint dependencies have been eliminated from the system.

5. Actuator-to-Cartesian Jacobian and Singularity Analysis

Velocity mapping uses the implicit differentiation of the loop equations:

$$\frac{\partial F}{\partial X}\, \dot{X} + \frac{\partial F}{\partial q_{\text{act}}}\, \dot{q}_{\text{act}} = 0$$

Defining $$q_{\text{act}} = [q_{13}, q_{23}, q_{33}, q_{4,2}]^T$$ and $X = [x, z, \theta, \psi]^T$ yields:

$$\dot{X} = J(X) \, \dot{q}_{\text{act}}, \qquad J(X) = -\left[\frac{\partial F}{\partial X}\right]^{-1} \left[\frac{\partial F}{\partial q_{\text{act}}}\right]$$

Partial derivatives for each leg, including the central leg, are given explicitly for calculating the $4 \times 4$ Jacobian blocks.

• Type-1 singularities: occur when $\det(\frac{\partial F}{\partial q_{\text{act}}})=0$, indicating loss of actuator-driven mobility in some passive motion directions.
• Type-2 singularities: occur when $\det(\frac{\partial F}{\partial X})=0$, so $J_a$ becomes ill-conditioned. Geometrically, type-2 cases arise when leg attachment points are collinear or cocircular in degenerate poses.

6. Model Assumptions and Grounding

The formulation assumes perfect rigidity of all links and joints, with friction only present at actuators. Universal and spherical joints are treated as ideal, introducing no constraints beyond their nominal axes. The manipulator is strictly planar: all motion is confined to the $X_F$-$Z_F$ vertical plane. Grounding is established via prismatic actuators and the R-U chain of the central leg, with fixed-base attachments aligned on a horizontal circle of radius $r$.

A plausible implication of these assumptions is that the model is ready to be implemented for applications requiring precise planar motion, with direct correspondence between control inputs and end-effector pose, supporting advanced control methodologies and singularity avoidance strategies (Valles et al., 2024).

7. Application Context: Lower Limb Rehabilitation

The 4-DoF model extends the minimum 3-DoF requirement for sagittal-plane rehabilitation, providing an additional degree that allows combinations of normal and tangential efforts, or direct torque application on the knee. This architecture fills a previously unaddressed gap for planar parallel manipulators capable of complex force and torque deployment in lower limb rehabilitation, with experimental results confirming tracking accuracy under the proposed control architecture (Valles et al., 2024).