Abstract
<title>Abstract</title> <p>This study investigates the balance and motion behavior of a two-wheel inverted pendulum robot (TWIPR) equipped with a scissor-pair control moment gyroscope (SPCMG) operating on inclined surfaces. In a conventional TWIPR, stability is achieved through continuous modulation of wheel torque. In contrast, the proposed architecture relies primarily on internally generated gyroscopic torque to regulate chassis posture. Because a CMG cannot sustain a constant torque in a fixed direction indefinitely, the robot must operate around a non-zero equilibrium tilt angle at which the gravitational component along the slope counterbalances the system dynamics.A complete nonlinear dynamic model of the robot is derived using the Euler–Lagrange formulation, incorporating gyroscopic coupling, chassis motion, and slope-induced gravitational effects. An analytical expression for the slope-dependent equilibrium configuration is obtained, revealing the relationship between terrain inclination and system structural parameters. To generalize the design insight, a dimensionless metric termed the Chassis-to-Robot Static Torque Ratio is introduced, linking mass distribution and geometric configuration to the equilibrium tilt angle. In combination with the surface friction coefficient, this parameter also defines the maximum admissible terrain inclination for stable operation.The proposed control framework employs two decoupled loops. Longitudinal motion is regulated using a proportional–integral–derivative (PID) controller acting on wheel torque, while balance stabilization is achieved through a linear quadratic regulator (LQR) controlling the gimbal dynamics. The controller maintains the equilibrium tilt while driving the gimbals toward zero precession, thereby preserving gyroscopic authority for disturbance rejection.The analytical model and control architecture are validated through high-fidelity multibody simulations implemented in MATLAB/Simulink. The results demonstrate strong agreement between analytical predictions and simulation outcomes, confirming the accuracy of the derived model and effectiveness of the proposed control strategy for gyro-stabilized two-wheel robots operating on inclined terrain.</p>