A Closed-Loop Dynamic Controller for Active Vibration Isolation Working on A Parallel Wheel-Legged Robot

The quadruped machine is the most agile-legged robot that can accomplish traversing by swing leg without high-complexity dynamics [1]. To improve the motion efficiency, a novel design of a mobile robot, called BIT-NAZA [2], that adopts arbitrary arrangements of legged and wheeled locomotion accounting for different types of terrains is introduced. The hybrid robot [3] tightly integrates the wheel on the end-effector of its foot. The wheel-leg favors parallel mechanism with inverted Stewart plat instead of classical three-joint serial structure, thereby inheriting existing benefit: The ability of the parallel wheel-leg to maintain the robot body horizontally with modest oscillation as soon as possible while the robot is rolling on uneven ground. To obtain a better vibration performance, the precise potion tracking loop for single wheel-leg is a basic procedure [4]. Developing a delicate force-driven controller based on dynamic model and merging adaptive impedance control loop is the primary scheme needed to achieve the single-foot vibration isolation [5]. It is also the first time to utilize these two control loops on the Stewart parallel mechanism to the best of the authors’ knowledge.

As a leg in legged locomotion, Stewart platform should swing along a trajectory to traverse the obstacle [6]. Apart from employing position control for Stewart manipulator [7] with conventional adaptive fuzzy sliding mode control method, Nabavi et al. [8] analyzed dynamic models using the Newton-Euler equation to form a closed-loop dynamic control techniques with force, thereby successfully tracking two trajectories. Incremental nonlinear dynamic inversion [9] addressed the implementation of a high-precision force controller for a Stewart hydraulic robot in the presence of parameter uncertainties, thereby validating the good performance of force and position tracking. There are also several position control approaches for robots with serial link [10], which tracks the target trajectories. A seven-link biped robot [11] coupled the feedforward compensator into fuzzy controller with inverse dynamic and noise reduction capability to track the desired motion. A new dynamic method [12] for fully reactive footstep planning characterized a careful feature of the planar spring-mass hopper. The LittleDog [13] robot employed QR decomposition to solve the floating-base inverse dynamics problem and predict force control, thereby increasing robustness despite of unknown perturbations.

Several algorithms are implemented to sustain a specific constatn posture or configuration for common serial-structure mechanism [14]. Five candidate Jitter cancelation algorithms are evaluated [15] for active vibration control on a spacecraft testbed, with adaptive linear model predictive control achieving one of the best disturbance rejection results with a 66\(\%\) overall amplitude reduction. A new cooperative model predictive control [16] is developed for Stewart stabilizable system. A new attitude balancing strategy [17] implemented an inverse kinematic scheme based on extended prediction self-adaptive control algorithm to generate full body motions that ensure the desired balancing performances for a hybrid wheel-legged mobility system. A design procedure for multi-strut vibration isolation platform [18] is the determination of optimal damping frequency for tree parameter isolator, which is tuned as the optimal damping frequency in three orthogonal directions. To obtain a favorable vibration performance, the dynamic model of the Stewart platform is constructed with flexible hinges [19] and coupled multiple flywheel system [20], forming state-space equations for control purposes. To suppress the self-excited vibration owing to flexibility, friction, backlash, coupling, and other nonlinear factors, a nonlinear controller and a fuzzy control algorithm [21] are designed to attenuate the self-excited vibration for the 3-RRR flexible parallel robot. The dynamic model of the Stewart platform is established by the frequency response function synthesis method. In the active control loop [22], the direct feedback of integrated forces is combined with the FxLMS-based adaptive feedback to dampen vibration. The finite frequency \(Hs_\infty\) vibration control technique and fractional-order \(PD_{v}\) are developed in a feedback loop [23] to suppress vibration modes and external disturbances for space flexible structures. However, there is no vibration isolation algorithm implemented on an inverted Stewart mechanism while the wheel fixed on end-effector is rolling on the rough terrain.

Additionally, a common method of disposing interaction with environment, such as vibration, is impedance control and force position hybrid control, which operates like an active spring [24]. A position controller for parallel robot [25] utilized the position impedance control to achieve smooth contact with environment, which is combined with Kalman filter to predict stability margin of robots inside zero moment point (ZMP) stability observer. The main application of variable impedance control is to adapt impedance parameters into position and velocity feedback [26], thus satisfying desired force value. A new adaptive impedance control [27] was proposed for force tracking that has the capability to track the dynamic desired force and compensate for uncertainties in environment, in terms of unknown geometrical and mechanical properties. The robotic manipulator [28] employed a novel adaptive impedance control, where the linear quadratic regulation (LQR) is formulated. The integral reinforcement learning is proposed to solve the given LQR with little information about the human arm model.

We present a robust control framework with force input based on dynamic model of Stewart wheel-leg to swing and isolate vibration for BIT-NAZA robot. The BIT-NAZA robot, a type of wheel-legged hybrid robot with Stewart parallel mechanism, is used to improve high payload and stability for vibration isolation. This paper mainly introduces the whole controller framework for single wheel-leg. The model predictive control based on dynamic model in inner-loop accomplishes precise position tracking. Upon position loop, the adaptive impedance control loop adjusts foot position to track target support force, thus keeps the robot body horizontal with ground. These parts constitute the closed-loop dynamic control framework, which is the first time the framework is applied to a parallel wheel-leg. The core of our work, which lies on the Newton-Euler dynamic equation, reveals the relationship between strut input force and foot configuration. To obtain a smooth output trajectory of wheel-legged mechanism, the desired velocity and control input increment are limited during each sampling instant. The feedback data is measured from displacement sensor installed in each strut and interactively, updating by forward kinematic calculation. Our methodology combined with adaptive impedance control in outer-loop, enables a sequence of corresponding control input for vibration isolation motion interacting with unknown environment, where the actual state feedback is also necessary to estimate environmental stiffness. For the availability validation of our proposed approach, the mentioned method is employed in two kinds of implements working on practical prototype, enforcing wheel-leg to track the swing trajectory, and driving the robot to pass through the speed bumps.

One of the most substantial benefits of parallel wheel-legged mechanism is active vibration isolation to decrease attitude variation of robot body while rolling on irregular terrain. As a result, the robot can acquire more robust position tracking performance while swinging foot and inner-loop isolating vibration. Our proposed algorithm is responsible for maintaining body horizontally according to data from the displacement and force transducer embedded in each strut. The wheeled speed consensus control [29] is not concerned, thereby leading robot to run at a constant default speed.

The block diagram for single-foot controller is demonstrated in Figure 1, where different boxes represent effective modules and procedure adjusting single Stewart configuration, \(i=1,\cdots ,4\). During the walking locomotion, robot swings foot to track obstacle-negotiating trajectory. We imposed the practical stretching length of strut as feedforward of model predictive controller and calculate the current configuration through Newton-Raphson iteration. The current configuration, velocity, and angular speed are combined into a dynamic model of Stewart wheel-leg, which forms a closed-loop position tracking controller with force input. The purpose of the vibration isolation is to alter the configuration of wheel-leg to adapt robot under the terrain variation. It also enables wheel-leg to track the desired force, sharing robot mass with other three wheel-legs, and ensuring close contact between the foot and the ground. When four wheel-legs stand on their median configuration, the body frame \(\mathcal {R}\), foot frame \(\mathcal {F}_i, i=1,\dots ,4\), and world frame \(\mathcal {W}\) are constructed and attached on robot as described in Figure 2. The proposed algorithm applies torque control to adjust the foot displacement along Z-axis, thereby tracking support force to achieve the vibration isolation. The most outstanding method to evaluate the performance of the algorithm is to measure the attitude variation of robot body and foot force along vertical direction. We illustrated the experimental data through operating the whole control framework on BIT-NAZA robot.

The cardan joints connect struts with moving and base platforms, ensuring zero constraint on six degrees-of-freedom (6-DOFs) for the moving platform. Progressively, invert the Stewart wheel-leg to describe dynamic model arrangement as depicted in Figure 3. The moving frame \(\mathcal {B}\) and base frame \(\mathcal {A}\) are fixed on moving and base plats respectively. Thus, the rotation matrix transformed from moving frame \(\mathcal {B}\) to base frame \(\mathcal {A}\) iswhere \(sX=\text{sin}X\), \(cX=\text{cos}X\), \(\phi _x\), \(\phi _y\), and \(\phi _z\) are the pitch, roll, and yaw angle of the moving platform, respectively. With position analysis in Figure 3, the vector loop equation of each linkage is acquired.

The dynamic model involves mathematical expression between the motion parameters of the moving platform, which includes configuration, velocity, acceleration, and the actuating force of each cylinder. Because of the complicated parallel mechanism with 6-DOFs, the dynamic model of Stewart presents a multi-degree-of-freedom, multi-variable, and high-nonlinear system. There are several approaches allowing multi-rigid systems, such as Stewart mechanism, to create dynamic models containing Newton-Euler, Lagrange, and Kane equations. This work adopts Newton-Euler equation to build the dynamic analysis and rapidly solves its solution in high-dimensional condition.

With aim at commanding the wheel-leg, the impression generated from rotational inertia of actuating cylinder emerges small compared to the mass of the robot body and the wheel group. Therefore, the inertia moment of electrical cylinder can be neglected when rotational and sliding friction of cardan joints and cylinders are ignored. The electrical cylinders are simplified as massless, which only afford tension and gravity along rod directions. According to force analysis from Newton’s second law, the motion equation related to center of mass of moving platform in base frame \(\mathcal {A}\) can be denoted as follows:where \(m_p\) is the mass of moving platform, \({{\varvec{g}}}\) is the gravity acceleration, \({}^{\mathcal {A}}{{\varvec{f}}}_i\) represents the driving force vector of electrical cylinder i, and \({}^{\mathcal {A}}\dot{{{\varvec{v}}}}_p\) indicates the acceleration of moving platform in base frame \(\mathcal {A}\).

Let \({}^{\mathcal {B}}{{\varvec{r}}}_i\) be the position vector of force action point in the moving frame \(\mathcal {B}\). On the basis of the applied moment on point \({}^\mathcal {B}{{\varvec{r}}}_i\) from actuating linkage i and generating rotational angular velocity \({}^\mathcal {B}{\varvec{\omega }}_p\), we compute the Euler rotation equation of moving platform in the moving frame \(\mathcal {B}\).The matrix formulation of rotational inertia tensor for moving platform in the moving frame \(\mathcal {B}\) can be calculated throughwhere inertia product along \(x^{\mathcal {B}}\), \(y^{\mathcal {B}}\), and \(z^{\mathcal {B}}\) axis in moving frame \(\mathcal {B}\) can be expressed as \(I_{xx}=\iiint _V(y^2+z^2)\rho \text{d}V\), \(I_{yy}=\iiint _V(x^2+z^2)\rho \text{d}V\), and \(I_{zz}=\iiint _V(y^2+x^2)\rho \text{d}V\) respectively, and inertia moment \(I_{xy}=\iiint _Vxy\rho \text{d}V\), \(I_{xz}=\iiint _Vxz\rho \text{d}V\), \(I_{yz}=\iiint _Vyz\rho \text{d}V\).

Execute coordinate conversion rotation matrix in Eq. (1) and get \({}^{\mathcal {B}}{{\varvec{R}}}_{\mathcal {A}}={{}^{\mathcal {A}}{{\varvec{R}}}_{\mathcal {B}}}^{\text{T}}\),Transform the Euler equation into base frame \(\mathcal {A}\), thus:The orthogonal matrix \({}^{\mathcal {A}}{{\varvec{R}}}_{\mathcal {B}}\) suffices \({}^{\mathcal {A}}{{\varvec{R}}}_{\mathcal {B}}{{}^{\mathcal {A}}{{\varvec{R}}}_{\mathcal {B}}}^ {\text{T}}={{\varvec{E}}}_I\), where \({{\varvec{E}}}_I\) is unit matrix,Substitute corresponding antisymmetric matrix \({}^{\mathcal {A}}{\varvec{\omega }}_p\times ={}^{\mathcal {A}}\widetilde{{\varvec{\omega }}}_p=\left[ \begin{matrix} 0&{}-{}^{\mathcal {A}}\omega _{p,z}&{}{}^{\mathcal {A}}\omega _{p,y}\\ {}^{\mathcal {A}}\omega _{p,z}&{}0&{}-{}^{\mathcal {A}}\omega _{p,x}\\ -{}^{\mathcal {A}}\omega _{p,y}&{}{}^{\mathcal {A}}\omega _{p,x}&{}0 \end{matrix}\right]\) and \({}^{\mathcal {A}}{{\varvec{b}}}_i\times ={}^{\mathcal {A}}\widetilde{{{\varvec{b}}}}_i=\left[ \begin{matrix} 0&{}-{}^{\mathcal {A}}b_{i,z}&{}{}^{\mathcal {A}}b_{i,y}\\ {}^{\mathcal {A}}b_{i,z}&{}0&{}-{}^{\mathcal {A}}b_{i,x}\\ -{}^{\mathcal {A}}b_{i,y}&{}{}^{\mathcal {A}}b_{i,x}&{}0 \end{matrix}\right]\) into Eq. (6), then,Multiply both sides with \({}^{\mathcal {A}}{{\varvec{R}}}_{\mathcal {B}}\), and signify driving force \({}^{\mathcal {A}}{{\varvec{f}}}_i={}^{\mathcal {A}}{{\varvec{s}}}_if_i\), where the scalar quantity \(f_i\) represents the value of driving forces.Set \({{\varvec{M}}}_p=\left[ \begin{matrix} m_p{{\varvec{E}}}_{3\times 3}&{}{{\varvec{0}}}_{3\times 3}\\ {{\varvec{0}}}_{3\times 3}&{}{}^{\mathcal {A}}{{\varvec{R}}}_{\mathcal {B}}{{\varvec{I}}}_p{{}^{\mathcal {A}}{{\varvec{R}}}_{\mathcal {B}}}^{\text{T}} \end{matrix} \right]\), \({{\varvec{C}}}_p=\left[ \begin{matrix} {{\varvec{0}}}_{3\times 3}&{}{{\varvec{0}}}_{3\times 3}\\ {{\varvec{0}}}_{3\times 3}&{}{}^{\mathcal {A}}\widetilde{{\varvec{\omega }}}_p{}^{\mathcal {A}}{{\varvec{R}}}_{\mathcal {B}}{{\varvec{I}}}_p{{}^{\mathcal {A}}{{\varvec{R}}}_{\mathcal {B}}}^{\text{T}} \end{matrix} \right]\), Jacobian matrix \({{\varvec{J}}}_p=\left[ \begin{matrix} {}^{\mathcal {A}}{{\varvec{s}}}_1&{}\cdots &{}{}^{\mathcal {A}}{{\varvec{s}}}_6\\ {}^{\mathcal {A}}\widetilde{{{\varvec{b}}}}_1{}^{\mathcal {A}}{{\varvec{s}}}_1&{}\cdots &{}{}^{\mathcal {A}}\widetilde{{{\varvec{b}}}}_6{}^{\mathcal {A}}{{\varvec{s}}}_6 \end{matrix} \right]\) and \({{\varvec{G}}}_p=\left[ \begin{matrix} {{\varvec{0}}}_{3\times 3}&{}{{\varvec{E}}}_{3\times 3}\\ {{\varvec{0}}}_{3\times 3}&{}{{\varvec{0}}}_{3\times 3} \end{matrix}\right]\). Then, the combined form of Newton motion and Euler rotation equation can be printed as follows:

Although the control input on Stewart is force, we still need configuration and force data of foot measured and calculated through sensors, forward kinematics, and Jacobian matrix as feedback. The purpose of MPC within the inner loop is to track the desired position and attitude. The outermost control loop exploits adaptive impedance control to track the desired support force, thereby maintaining whole body stability during rolling process.

To obtain better performance of the proposed controller, a co-simulation system is constructed by combining the MATLAB/SIMULINK and ADAMS working on a single Stewart mechanism, as demonstrated in Figure 3. The model parameter setting is listed in Table 1. The design of dangling moving plat can exclude the influence force from ground to a robot foot that clearly illustrates the position tracking performance of the controller. Similarly, the force and displacement sensors internally embedded in each strut measure its load-carring force and expansion amount, thereby serving as system feedback and solving the force burden and configuration variation of moving base through the Jacobian matrix. The step and sinusoid are taken as trajectory input to adjust the parameter optimization of the controller, in which the sample duration \(T_s\) is 0.0005 s.

Progressively, the traditional control variate technique is adopted to mutually schedule parameters containing predictive horizon \(N_p\) and control horizon \(N_c\). Given step signal amplitude 0.15 (it can be set as displacement or angle along X-, Y-, or Z-axis, it is tested as X-axis displacement in this case) and input constraint within ±700 N, evaluate the tracking performance of model predictive control (MPC) for step signal under different \(N_p\) and \(N_c\).

Comparing Figure 4 with Figure 5, we observe that the variation of predictive horizon \(N_p\) has a smaller impression on the system step response. Nevertheless, the control horizon \(N_c\) which differs from sample duration and stands for frequency how long import the optimal control sequence into the system, has a significant impact on the step response. Table 2 quantizes the step response for different \(N_c\).

The updating frequency of optimal control sequence decreases with an increase in \(N_c\), thereby reducing rising time and steady-state relative error. Additionally, in view of the large calculation amount and weakening real-time computing capacity originated from decreasing \(N_c\), control horizon of the proposed controller is set as \(N_c=2\).

For the parameter of gain/integration (G/s) module depicted in Figure 1, the gain and integral factor also need to be turned. We define a sinusoidal signal with a frequency of 2 Hz and an amplitude of 0.15 rad as pitch angle reference. The corresponding response of the control system does not produce shock while increasing the gain value P, as shown in Figure 6. With the fixed predictive horizon \(N_p=5\) and control horizon \(N_c=2\), the amplitude error of sinusoidal response is 0.0004 rad with 0.0005 s delaying time when we set the gain of G/s module as \(P=0.12\), thus satisfying tracking precision and requirement of dynamic property.

After the parameter adjustment step, our experiment with the optimal control force input exposes an interesting challenge with the effect from environment. Based on the practical requirements of the BIT-NAZA robot, both a Bezier trajectory for the corresponding swing leg in legged locomotion and vibration isolation in wheeled motion are examined on physical prototype. The displacement and force are recorded from corresponding sensors embedded in electrical cylinders to calculate the corporate displacement and force for wheel-leg, whose update time is the same as practical control duration 3 ms.

We presented a novel control system for a Stewart parallel hybrid wheel-legged robot. The robot mainly contains two tracking loops. The objective function based on the dynamic model of Stewart mechanism, solved optimal control sequence and iteratively fresh state during the predictive horizon to complete inner-loop position control. Furthermore, the function showed how the parameter of our controller can be leveraged to significantly improve the control accuracy. This, improved accuracy allows users to make a tradeoff between decreasing calculation amount and enhancing precision. To better adapt to unknown terrain, adaptive impedance control in outer loop is implemented into this structure, thereby estimating environmental stiffness and combining with other impedance parameters of wheel-leg. We demonstrated the effectiveness of our controller by tracking Bezier trajectory for swinging foot within single inner-loop controller and vibration isolation during rolling motion using intact control framework, all of which were tested on a wheel-legged hybrid robot. We observed that speed bump passing time of adaptive impedance control is shorter than that of constant impedance control. Besides, the force-tracking fluctuation and postural variation became smaller. Adaptive impedance quickly adjusts foot position to adapt in terrain variation, thereby decreasing resistance force along the horizontal direction. The proposed controller can force wheel-leg to swing in legged locomotion and keep robot body horizontal with ground in wheeled motion.

The coupled displacement adjustment from impedance control when all wheel-legs are equipped with the proposed controller. This situation still leads to additional body vibration, even though only two wheel-legs receive command from the proposed controller. The further vibration isolation framework requires significant improvement to decrease coupling of four wheel-legs, where the auxiliary event trigger regulator based on postural variation is efficient. As for the application in future industry, the nominate algorithm can help wheel-legged robot to rolling on rough terrain. The operating field contains the resource exploration, disaster relief, and package transportation.