The design of humanoid robots has seen significant advancements in recent decades, with a key focus on enhancing their operational capabilities in complex, human-centric environments. A critical differentiator for a humanoid robot is its dual-arm configuration, which offers substantial advantages over single-arm systems by enabling the execution of sophisticated bimanual tasks through independent or synchronous operation. To ensure this versatility, humanoid robots often employ redundant 7-degree-of-freedom (DOF) manipulators, which help avoid singularities in specific workspace regions. The effective workspace of these dual arms—the spatial region within which the robot can meaningfully operate—is paramount, directly influencing the scope of tasks a humanoid robot can perform, from industrial assembly and logistics to healthcare assistance.
In this research, we present a targeted geometric analysis method to determine the optimal dual-arm workspace for a humanoid robot in a logistics handling scenario, specifically for box-carrying tasks. Unlike probabilistic methods like Monte Carlo simulations, which generate point clouds of reachable end-effector positions but struggle to define the coordinated workspace for two arms under task constraints, our approach uses deterministic geometric modeling. We establish explicit mathematical relationships between the design parameters of the dual-arm mounting base, the kinematic configuration of the manipulators, and the dimensions of the target object. By doing so, we can calculate the base parameters that maximize the usable workspace for a given task, providing a clear design guideline for the torso structure of a humanoid robot.
The operational potential of a humanoid robot is intrinsically linked to its physical architecture. The base structure to which the arms are attached is not merely a support; it is a fundamental design variable that critically impacts the robot’s stability, load capacity, and most importantly, the spatial envelope for cooperative arm movement. Key parameters of this base include the installation width $D_b$ (the lateral separation between the arm mounting points) and the installation angle $\theta$ (the forward tilt of the arm’s first link relative to the vertical torso axis). The primary challenge we address is: given a specific humanoid robot arm model and a range of target box sizes, how do we select $D_b$ and $\theta$ to maximize the cooperative workspace for stable, collision-free box handling?
Geometric Modeling of the Humanoid Robot Dual-Arm System
Our analysis begins with the kinematic modeling of a standard 7-DOF redundant manipulator, representative of arms used in advanced humanoid robots. The Denavit-Hartenberg (D-H) convention is applied to establish coordinate frames for each joint, from the base (Frame 0) to the end-effector (Frame 7). The constant D-H parameters for our manipulator model are summarized in Table 1, where $a_{i-1}$, $\alpha_{i-1}$, and $d_{i-1}$ are the link length, link twist, and link offset, respectively, and $\theta_i$ is the variable joint angle.
| Joint $i$ | $a_{i-1}$ (mm) | $\alpha_{i-1}$ (°) | $d_{i-1}$ (mm) | Variable $\theta_i$ | Range (°) |
|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | $\theta_1$ | (-178, 178) |
| 2 | 0 | 90 | 102 | $\theta_2$ | (-130, 130) |
| 3 | 0 | -90 | 152.5 | $\theta_3$ | (-178, 178) |
| 4 | 0 | 90 | 103.5 | $\theta_4$ | (-135, 135) |
| 5 | 0 | -90 | 128.5 | $\theta_5$ | (-178, 178) |
| 6 | 0 | 90 | 81.5 | $\theta_6$ | (-128, 128) |
| 7 | 0 | -90 | 115 | $\theta_7$ | (-360, 360) |
For the purpose of workspace boundary analysis related to box handling, we simplify the model by focusing on the joints and links that primarily govern the gross positioning of the end-effector in the sagittal and coronal planes. We define the key link lengths for our geometric analysis as follows: $l_1$ (from shoulder joint to elbow joint axis), $l_2$ (upper arm link), $l_3$ (forearm link), and $l_4$ (wrist to end-effector flange). An additional length $l_{tool}$ represents the gripper or hand. The combined system of two such arms mounted on a base defines our humanoid robot torso assembly.
Analysis of Cooperative Motion Limit Positions
The coordinated workspace for box handling is bounded by extreme positions where the arms are fully extended or at risk of self-collision. We analyze these limits in the lateral (left/right) and sagittal (forward/backward) directions, considering two main constraint categories: tool-frame constraints and base-frame constraints.
Tool-Frame Constraint: This is defined by the handling width $D_w$, which is the fixed distance the two end-effectors must maintain to grasp opposite sides of a box stably.
Base-Frame Constraints: These are defined by the base design parameters: the installation width $D_b$ and the installation angle $\theta$.
Lateral (Left/Right) Limit Analysis
When the humanoid robot attempts to move a box to its leftmost or rightmost position, the limiting factor is either the maximum extension of one arm or the collision between the links of the opposite arm. For a symmetric analysis, we consider the leftward motion. The left arm’s joint $\angle L_2$ (simplified representation of the major elevation joint) can theoretically reach 180° for maximum extension. Simultaneously, the right arm’s corresponding joint $R_2$ must not decrease below a minimum collision-safe angle $\theta_m$, a parameter intrinsic to the arm’s mechanical design.
We identify a critical handling width $D_{w}^{T}$ that marks the transition between two limiting regimes:
- Collision-Limited Regime ($D_w < D_{w}^{T}$): The right arm hits its collision limit ($R_2 = \theta_m$) before the left arm reaches full extension. The leftward extreme distance $LD_m$ is determined by $\theta_m$.
- Extension-Limited Regime ($D_w > D_{w}^{T}$): The left arm reaches full extension ($\angle L_2 = 180^\circ$) before the right arm collides. Here, $LD_m$ is determined by the 180° limit of $\angle L_2$.
To find $D_{w}^{T}$, we analyze the transitional configuration where $\angle L_2 = 180^\circ$ and $R_2 = \theta_m$ simultaneously. Introducing auxiliary angles $\theta_1$ and $\theta_2$ based on the arm geometry, we derive the following relationships:
$$ D_{w}^{T} = D_b + (2l_1 + l_2)\cos\theta – (l_2 + l_3)\sin\theta_2 – l_3\sin\theta_1 $$
$$ (l_2 + l_3)\cos\theta_2 – l_3\cos\theta_1 = l_2\sin\theta $$
$$ \theta_2 = 90^\circ – (\theta_m – \theta) $$
Solving this system yields $D_{w}^{T}$. Subsequently, the leftward limit distance $LD_m$ for any given $D_w$ can be calculated piecewise:
For $D_w < D_{w}^{T}$:
$$ LD_m = (l_2 + l_3)\cos(\theta_m – \theta) + \frac{D_w}{2} – l_1\cos\theta – \frac{D_b}{2} $$
For $D_w > D_{w}^{T}$:
$$ LD_m = (l_1 + l_2)\cos\theta – l_3\sin\theta_1 + \frac{D_b}{2} – \frac{D_w}{2} $$
where $\theta_1$ and $\theta_2$ are found by solving the system derived from the geometry with $\angle L_2 = 180^\circ$:
$$ A = D_b – D_w + (2l_1 + l_2)\cos\theta $$
$$ B = l_2\sin\theta $$
$$ \theta_1 = -2\arctan\left(\frac{2A l_3 \pm \sqrt{(A^2+B^2-l_2^2)(-A^2-B^2+l_2^2+4l_2l_3+4l_3^2)}}{-A^2-B^2+2l_3B+l_2^2+2l_2l_3}\right) $$
A similar derivation provides the rightward limit distance $RD_m$, and by symmetry, $|LD_m| = |RD_m|$ for a centered workspace.
Sagittal (Forward/Backward) Limit Analysis
For forward limits, the arms typically move symmetrically. The forward extreme distance $FD_m$ (from base to the front of the gripped box) is also subject to the link collision constraint $\theta_m$. The critical width $D_{w}^{T,f}$ separating collision-limited and extension-limited forward motion is:
$$ D_{w}^{T,f} = D_b + 2l_1\cos\theta – 2(l_2 + l_3)\cos(\theta_m – \theta) $$
For the common case where $D_w > D_{w}^{T,f}$ (extension-limited), the forward limit is given by:
$$ FD_m = \sqrt{(l_2+l_3)^2 – \left(l_1\cos\theta + \frac{D_b}{2} – \frac{D_w}{2}\right)^2} + l_1\sin\theta + l_4 + l_{tool} $$
Analyzing the backward limit $BD_m$ is more complex due to potential collisions between different links (e.g., forearm with upper arm). The limiting condition could be either joint $R_3$ or $R_4$ reaching $\theta_m$ first. We analyze both scenarios by constructing geometric relationships. If $R_3$ is limiting, we define auxiliary lengths and angles:
$$ l_p = \frac{2l_1\cos\theta + D_b – D_w}{2} $$
$$ l_k = \sqrt{l_2^2 + l_3^2 – 2l_2 l_3 \cos \theta_m} $$
$$ l_n = \sqrt{l_k^2 – l_p^2} $$
Then, $BD_m = l_n + l_4 + l_{tool}$.
If $R_4$ is limiting, we solve for the corresponding $R_3$ using the condition $\theta_k + \theta_n = \pi – \theta_m$ and the laws of sines and cosines in the relevant triangles, subsequently calculating $BD_m$. The smaller $BD_m$ value from the two possible limiting conditions defines the actual backward workspace boundary.
Experimental Parameter Optimization and Validation
We applied this geometric analysis method to a specific humanoid robot arm model with the following parameters: $l_1=242$ mm, $l_2=256$ mm, $l_3=210$ mm, $l_4=144$ mm, $l_{tool}=100$ mm, and $\theta_m = 50^\circ$. For a target handling width of $D_w = 300$ mm, we sought the optimal base parameters $D_b$ and $\theta$.
Using MATLAB, we generated mapping relationships between the base parameters and the extreme distances ($LD_m$, $RD_m$, $FD_m$, $BD_m$). The analysis, constrained to a reasonable design space of $0 < \theta < 50^\circ$ and $0 < D_b < 500$ mm, yielded the following insights:
- Lateral Workspace: The maps for $LD_m$ and $RD_m$ were symmetric. Smaller values of $D_b$ consistently yielded larger lateral workspaces. For a given $D_b$, an optimal $\theta$ existed to maximize lateral reach.
- Forward Workspace: Both $D_b$ and $\theta$ positively influenced $FD_m$, with $\theta$ having a more pronounced effect. Larger angles $\theta$ significantly increased forward reach.
- Backward Workspace: A smaller $BD_m$ is desirable as it indicates the arm can pull an object closer to the torso. $D_b$ was the dominant factor affecting $BD_m$, with smaller $D_b$ leading to smaller $BD_m$ (i.e., larger backward workspace).
The primary design trade-off became clear: minimizing $D_b$ maximizes the lateral and backward workspace but must be balanced against the need for structural stability and assembly feasibility. A very small $D_b$ could also lead to physical interference between the arm drive systems inside the torso. For our humanoid robot platform, we selected a minimum practical $D_b = 100$ mm. With $D_b$ fixed, the mapping indicated that a base angle $\theta$ in the range of $15^\circ$ to $20^\circ$ provided an excellent compromise, maximizing the lateral workspace while maintaining substantial forward reach. We chose $\theta = 15^\circ$ for our final design.
To validate the mapping relationships, we conducted virtual experiments in simulation software, varying $D_b$ and $\theta$ while keeping $D_w$ and $\theta_m$ constant. The measured extreme distances closely matched the values predicted by our geometric formulas. Sample results for the forward and lateral workspaces are shown in Tables 2 and 3, confirming the accuracy of our model.
| $\theta$ (°) | $D_b$ (mm) | $FD_m$ (mm) |
|---|---|---|
| 10 | 100 | 731.02 |
| 15 | 100 | 753.03 |
| 30 | 100 | 817.93 |
| 15 | 150 | 744.76 |
| 15 | 200 | 734.88 |
| 15 | 300 | 709.77 |
| $\theta$ (°) | $D_b$ (mm) | $LD_m$ / $RD_m$ (mm) |
|---|---|---|
| 10 | 100 | 218.65 |
| 15 | 100 | 247.97 |
| 30 | 100 | 234.44 |
| 15 | 150 | 222.97 |
| 15 | 200 | 197.97 |
| 15 | 300 | 147.97 |
Conclusion and Discussion
We have developed and demonstrated a geometric constraint-based method for analyzing and optimizing the dual-arm cooperative workspace of a humanoid robot for specific tasks like box handling. By explicitly modeling the interactions between the arm kinematics, the target object size ($D_w$), the mechanical collision limit ($\theta_m$), and the torso base parameters ($D_b$, $\theta$), we can derive closed-form mathematical relationships that define the workspace boundaries. This approach provides direct, intuitive design guidance that is superior to iterative trial-and-error or purely numerical sampling techniques.
Our analysis for a specific logistics scenario revealed that a smaller base width $D_b$ is generally beneficial for maximizing the lateral and backward workspace, while a moderate forward tilt angle $\theta$ (e.g., $15^\circ-20^\circ$) effectively increases the forward reach without severely compromising other dimensions. The chosen optimal parameters ($D_b = 100$ mm, $\theta = 15^\circ$) for our humanoid robot represent a balance between maximizing operational space and ensuring mechanical stability and practicality.
The significance of this work lies in providing a foundational design framework. For any given humanoid robot arm model and a defined task envelope, designers can use this methodology to calculate the optimal torso base geometry upfront, thereby ensuring the robot is physically capable of performing its intended functions within the largest possible workspace. This is a crucial step in the efficient design of capable and versatile humanoid robots. Future work will involve extending this geometric analysis to incorporate dynamic constraints, obstacle avoidance in the workspace, and the optimization of base parameters for a broader set of complex dual-arm manipulation tasks.