Underactuated tendon-driven hands offer compact actuation and passive compliance, but tendon elongation under restoring-spring loading introduces configuration-dependent joint deviations. This paper presents H-PAC, a modular 6-actuator, 15-DoF robotic hand with a control-oriented modeling and implementation framework. A sparse analytical actuator-joint model is derived from the tendon-routing geometry, and a mechanics-based compensation model is developed to account for tendon-elasticity-induced joint errors. The proposed method is implemented in a hierarchical architecture: a host computer performs workspace-constrained posture mapping and compensation, while an ESP32 generates synchronized commands for six position-controlled servos. The same control parameters and execution strategy are used across all tasks without task-specific retuning. Monotonic servo-sweep experiments show that the compensation substantially improves joint-angle prediction. The MAE of the index DIP joint decreases from 1.15 degrees to 0.18 degrees, and all nine evaluated joints achieve an MAE below 0.23 degrees. Representative postures and grasping configurations are further executed using the same control pipeline without external joint or force sensing in the control loop. The results demonstrate a practical approach to improving posture reproducibility in compact underactuated robotic end-effectors.
Figures & tables
Fig. 1: Overview and DOF mapping of H-PAC Hand. (a) Human hand annotated with anatomical DOFs (MCP/PIP/DIP, MP/IP, and CMC). (b) The H-PAC Hand with 15 joint DOFs q1−q15 : four long digits each with MCP/PIP/DIP flexion–extension, and the thumb with IP/MP flexion–extension and CMC abduction–adduction.
Fig. 2: Tendon routing and analytic modeling for four fingers. Four-finger figure-eight routing from full extension to an intermediate flexion: straight segments Lk , joint pulleys of radius Ri , cables connected to the servo hubs (purple), the passive ”cable-spring” return path (green), parameters related to the closed-form geometric expression of the MCP joint ( α , d1 , d2 ), and joint angles qMCP , qPIP , qDIP . qi,ref (where i={DIP,PIP,MCP} ) denotes the structural geometric limits of the independent finger module for analytical derivation, while the actual operational range limits are constrained by the palm housing to prevent excessive flexion.
Fig. 3: Tendon routing and analytic modeling for thumb. The thumb uses figure-eight tendon routing for coupled MP/IP flexion, parameterized by αth , d1,th , and d2,th , with joint angles qMP and qIP . The structural limits qi,ref ( i={IP,MP} ) are defined analogously to the four-finger model. Servo-6 drives the CMC joint, with its output axis coaxial to the CMC joint axis and the servo horn directly rigidly attached to the CMC joint.
Finger Joint
Joint Label
Admissible Workspace [qmin,qmax] (rad)
Thumb IP
q1
[0.00,0.99]
Thumb MP
q2
[0.00,1.25]
Index DIP
q3
[0.00,1.31]
Index PIP
q4
[0.00,1.61]
Index MCP
q5
[0.00,1.27]
Middle DIP
q6
[0.00,1.28]
TABLE I: Admissible workspace of each finger joint of H-PAC. A monotonic servo sweep dedicated to workspace determination is used to obtain [qmin,qmax] .
Fig. 4: Comparative analysis of joint angle residuals under uncompensated and compensated control. This figure illustrates the relationship between the angle residuals and the servo control angles for target joints across the five fingers. Red circles represent the original uncompensated residuals (Residual_ideal) and green squares denote the residuals after mechanics-based compensation (Residual_comp). All subplots are standardized with a Y-axis range from −2.0 to +2.0 degrees to facilitate a direct comparison of performance.
Fig. 5: Mean Absolute Error (MAE) comparison of joint angles for five fingers of H-PAC Hand under uncompensated and compensated conditions. The red bars represent the MAE between real joint angles and ideal joint angles (without compensation), while the green bars denote the MAE between real joint angles and compensation joint angles (with compensation). The fingers include the thumb (with MP and IP joints) and four fingers (each with MCP, PIP and DIP joints). The MAE is calculated in the unit of degrees ( ∘ ).
Fig. 6: Representative posture and grasping demonstrations of the H-PAC Hand. Left: Hand gestures, including precise thumb-to-finger opposition (thumb opposing the index, middle, and ring fingers respectively), representative multi-finger combinations and digit gestures 1-8. Right: An extended set of 16 representative grasps under the same controller, covering diverse contact topologies (enveloping, lateral, and tip pinches) and tool morphologies (rod-like, ring-like, and press-type). All results from the same control stack (analytic mapping + mechanics-based compensation), without external joint or force sensing in the control loop.
All authors are with the School of Engineering Mathematics and Technology, and Bristol Robotics Laboratory, University of Bristol, Bristol, BS8 1UB, U.K.