Environmental Adaptive Control of a Snake-like Robot With Variable Stiffness Actuators

Dong Zhang,, Hao Yuan, and Zhengcai Cao,

Abstract—This work investigates adaptive stiffness control and motion optimization of a snake-like robot with variable stiffness actuators. The robot can vary its stiffness by controlling magnetorheological fluid (MRF) around actuators. In order to improve the robot’s physical stability in complex environments, this work proposes an adaptive stiffness control strategy. This strategy is also useful for the robot to avoid disturbing caused by emergency situations such as collisions. In addition, to obtain optimal stiffness and reduce energy consumption, both torques of actuators and stiffness of the MRF braker are considered and optimized by using an evolutionary optimization algorithm. Simulations and experiments are conducted to verify the proposed adaptive stiffness control and optimization methods for a variable stiffness snake-like robots.

I. Introduction

SNAKE-LIKE robots could move flexible in complex environments due to their multiple degree of freedoms(DOFs) and various gaits [1]. Since the first snake robot was designed by Ohnoet al. [2], some snake robots have been developed. Snake robots with different mechanisms show variable characteristics. Those equipped with passive wheels and active joints can move smoothly in flat terrains [3]. Those composed of active wheels and passive joints can move fast in rough terrains [4]. Those constructed by active joints and active wheels can move fast and efficient on flat grounds and slops [5]. Those formed by joints that can bend and elongate can be used in rough terrains and narrow spaces due to their flexibilities [6].

Based on researches of mechanism designing, some gaits including planar and 3-D gaits are investigated. In [7]–[10]planar gaits are researched and optimized. In [11]–[13], some 3-D gaits are designed and researched. Built on these researches, they are used to execute special tasks such as disaster rescuing [14], [15]. However, they cannot change their stiffness like their biology equivalents.

Since most snake robots are driven by motors and reduction gears, it is difficult to control their stiffness directly [16]. To solve this problem, some flexible mechanisms are researched in the past years [17]–[20]. In [17], a spring parallel variable stiffness actuator is designed and analysed. In [18], a snake robot with flexible backbones is proposed to get changing stiffness. In this robot, modules are driven by ropes and it has advantages of parallel and series mechanisms. In [19], a snake-like robot with variable stiffness actuators for searching and rescuing tasks is presented. With the development of material science, many new materials are used on robots, such as magneto-rheological fluid (MRF). In [20], a disk-type MRF brake for a mid-sized motorcycle to decelerate is designed.Those mechanisms are equipped with variable stiffness actuators but they cannot transit their stiffness according to environments adaptively.

For variable stiffness snake-like robots, their environmental adaptability and stiffness control have attracted many attention in recent years [21]. In [22], a virtual shape based compliant method is proposed to control a robot to move smoothly in an environment with some obstacles. In [23], a modular robot is controlled to change its body passively to realize compliant locomotion on horizontal pipes with different radii. In [24], a snake robot composed of several parallel elastic variable stiffness actuators with springs is designed. Stiffness of its springs is controlled to minimize torques of its actuators. Although stiffness of actuators can be controlled, other performances such as energy consumption are not considered.

To avoid stiff impulses of snake robots, an improved central pattern generator is proposed in [25]. In order to analyze the relationships between the robot’s performances and stiffness coefficients of the flexible component, simulations about the effects of compliant intervertebral discs are carried out in [26].To balance moving capability and computation cost of snakelike robots, a joint-level torque feedback control method is proposed to transit their shapes in [27]. Above methods could passively control stiffness of snake-like robots. Less attention have been paid on active stiffness control.

This work analyzes the dynamic of the variable stiffness actuator which was designed in our previous work [28]. To improve the stability and environmental adaptability of the robot, its stiffness is controlled actively and optimized.Besides stiffness of actuators, the energy consumption is also taken into account. A multi-objective evolutionary algorithm is used to optimize energy consumption and stiffness configuration. Stiffness of the robot is configured to avoid negative disturbance of emergency situations such as collisions.

The remainder of this work is organized as follows: the variable stiffness mechanism and performances of a snakelike robot are described in Section II. The dynamic model and compliant control method based on stiffness-energy optimization are proposed in Section III. Simulation and experiments are conducted to verify the proposed control methods in Section IV. Finally, conclusions are given in Section V.

II. The Variable Stiffness Snake-like Robot

To realize variable stiffness of a snake robot, this work investigates a joint with a magneto-rheological fluid braker(MFB) proposed in our previous work [28]. As shown in the right part of Fig. 1, the MFB mainly contains an MRF, a disk,a coil, a shell and a rotary axis. The shear yield stress of the MFB can be controlled by changing magnetic fields around it.As shown in Fig. 1, the MFB is fixed in the module’s upper end to generate a damping torque. The stiffness can be controlled by the torque. When the coil is energized, a magnetic field will be generated around the MRF (the red line in Fig. 1). The magnetic field affects states of the MRF and the yield damping torque. The damping torque Γ can be calculated as [18]

where ω is the angular velocity of the disk,Iis the electric current on the coil, τy(I) is the shear yield stress that depends on the magnetic flux density of the MRF,his the thickness of the MRF and it is set as 1 mm in this work, η is a constant dependent of the MRF,Randrare the external radius and the internal radius of the MRF, respectively. As shown in Fig. 2,the direction of the damping torque is reverse to the direction of ω.

Fig. 1. The snake-like robot with variable stiffness actuators. The left side is the joint and the right side is the structural section view of the MFB.

In this work, we make a prototype module for a variable stiffness snake-like robot. To get the relationship between MFB’s torque and voltage, several experiments are conducted.Torques generated by different voltages are measured by using a torque meter. As shown in Fig. 3, the actual damping torque of a variable stiffness joint could reach 0.7 N·m.

Fig. 2. Generation of the damping torque from a top side view of the MFB.

Fig. 3. Damping stiffness performance of the prototype joint.

III. Adaptive Control of a Snake-like Robot

In this section, a dynamic model of the snake-like robot is derived. An environmental adaptive control law combines a stiffness-energy optimization algorithm is designed. Then,changes of stiffness and drive torques of the control law are analyzed.

A. Kinematics of the Snake-like Robot

As shown in Fig. 4, ann-link snake robot is connected byn?1 joints. Each link has a same massmand a same length 2l.The center of mass (CM) of a link is assumed to locate at its center point (xi,yi) . Position of the head is (xh,yh). The robot is propelled forward by swinging its links continuously. The viscous friction model is used to compute the frictions in this work. Viscous frictions are assumed to act on CMs of links.We make use of mathematical symbols described in Table I.

The global frame position of the robot’s CM is calculated as

wheree=[1,1,...,1]T∈Rn.

According to motion characteristics of snake-like robots,linksiandi+1 must comply the following constraints:

Fig. 4. Schematic diagram of a snake-like robot in the global coordinate.

TABLE I Mathematical Symbols

Constraints of all links are

where

Velocities of all links can be computed as:

whereSθ=diag{sinθ},Cθ=diag{cosθ} andK=AT(DDT)?1D.

B. Dynamics of the Snake-like Robot

Similarly to [29], the dynamic model of a snake robot is derived in this work. As shown in Fig. 4, the configuration space can be defined asLetdenote the translational velocity of the head, other notations are defined the same as [29].

Dynamic model of the snake-like robot can be written as

where

As shown in Fig. 5, a framework is designed for the proposed controller. When the snake-like robot is subjected to an external force, the stiffness equation is solved according to the stiffness model and relevant motion parameters. The energy consumption can be computed by the dynamic model.To improve the environmental adaptability of the robot, an optimization algorithm is used to optimize its stiffness and torques.

The stiffness of the snake-like robot is helpful to avoid deformations under external forces. Snake robots have a characteristic that links follow the motion of the head link rhythmically. In addition, the head link is equipped with many sensors to measure the environment while the robot executing tasks. Therefore, both the stiffness of joints and the head link need to be optimized. In order to solve this problem, a control framework which combines the dynamic model and a stiffness-energy optimization algorithm is designed.

Fig. 5. The proposed control framework of the snake-like robot.

The force Jacobi matrix is the transpose of motion Jacobi matrix τ =JT(q)f. Therefore

where

fis the sum of force applied on the head link,J(q) is the Jacobi matrix,Kθis the stiffness matrix,kiis the stiffness of jointiandKcis the collaborative matrix that compensates the variable stiffness caused by rotations of joints.

D. Input Torques of Joints

The body shape of a snake-like robot can be approximated by a serpenoid curve. The serpenoid curve is characterized as

wheresis the arc-length coordinate of a point, ρ(t,s) is the curvature at points,Tdecides the number of periods ins∈[0,1], α and ωhare angle and frequency of the curve, respectively. To achieve this body shape, reference yaw joint angles are

whereNhis the number ofSshape in the horizontal plane, and ?0is the phase offset of yaw joints. Fori= 1, we define ?0=0. An exponentially stable joint controller is used to make the joint angles track the reference angles. New inputs can be computed as

wherekd>0 andkp>0 are gains. The torques include two parts, the first one is generated by motors and the second one is generated by variable stiffness actuators.

E. Optimization of Torques and Stiffness

In the control system, an exponentially stable joint controller and a dynamic model are used to solve initial torquesThe optimal torquescan be obtained by using an optimization algorithm. Compared with their initial states, the stiffness of the head link and all joints increases obviously,which will be verified in the next section.

In this work, the optimization problem is separated into two functions. The first one composes energy consumptions of a motor and a coil. It is a function of motor’s torque and electric current

The second one is the stiffness of the head link

It can be solved by using the stiffness model and parameters such as θiand τmi.

To optimize the stiffness and energy consumption, a multiobjective optimization algorithm should be used. In the past years, several optimization algorithms are researched [30]–[32].In this work, a multi-objective evolutionary algorithm based on decomposition in [30] is used to optimize the stiffness and energy consumption of the snake-like robot. The optimization algorithm is shown in Fig. 6. The objects of the optimization can be described as minimizingf1and maximizingf2. The constrained space is limited by MFB’s damping torque as shown in Fig. 3. The multi-objective optimization problem is separated into two single-objective optimization problems by using this algorithm. These sub-problems are optimized by using an evolutionary algorithm based on the information of a certain number of adjacent problems.

Fig. 6. Flowchart of the multi-objective optimization algorithm.

Since the objective functions are mutually exclusive, it is valuable to solve the equilibrium solutions of the objective functions instead of their optimal values. The equilibrium solutions are defined as Pareto optimal values. In the decomposition-based multi-objective optimization, the Chebyshev decomposition method is used to divide the problem intoNscalar optimization subproblems

wherex∈?, ? is a decision space,z=(z1,...,zm) infers to the reference point andThe coordinate of the minimum value of each target component [λ1,...,λn]Tis a uniform distributed weight vector

IV. Evaluation

To test performances of the proposed adaptive stiffness control and optimization methods for a variable stiffness snake robot, we have conducted extensive simulations in MATLAB and ADAMS by using parameters in Table II.

The snake robot’s model is actualized and simulated in MATLAB R2016b. When the robot moves in lateral undulation gait with a velocity 0.1 m/s, its displacement and joint angles change rhythmically as shown in Fig. 7. Joints swing rhythmically after a transition period.

Simulations are conducted in Matlab to test the optimization algorithm proposed in Section III. In these simulations, robots are disturbed by external forces. Their torques and stiffness are solved by using the proposed optimization algorithm.Results in Figs. 8(a)?(b) are robot’s stiffness in thexand theydirections. It is clear that robot’s stiffness change periodically.Results in Figs. 8(c)?(d) are changes of torques of the first and fourth joints. It is seen that joints’ torques increase with their stiffness. It can be concluded that the torque of each joint changes little but the stiffness in thexandydirections increases obviously. From these results, it can be obtained that the proposed adaptive stiffness control method is useful to improve the robot’s stability.

To test the stability of a robot with increased stiffness,simulations are conducted under external collisions. The stability is assessed by the offset of the robot’s displacement.We conducted two groups of simulations in ADAMS as shown in Fig. 9. In the first group of simulations, the robot’s stiffness keeps constant. In the second group of simulations,the robot adjusts its stiffness by using methods proposed in Section III. In these simulations, the impactor collides the second joint of the snake-like robot at 0.7 s. The object detaches the second joint and begins to contact the first joint at 1.4 s. The impactor moves away from the robot at 2.4 s.Displacements of snake robots are shown in Fig. 10. It is seen that the displacement of a robot with variable stiffness is less than the displacement of another robot with constant stiffness.It is concluded that the proposed adaptive stiffness control methods can improve the robot’s stability when the robot is hit by impactors.

To verify the proposed control methods, experiments of a snake-like robot are conducted as shown in Figs. 11 and 12.The 5-module prototype is connected by 4 joints. Each joint of the robot is driven by a 42-step motor. Each joint is equipped with a variable stiffness element to control its stiffness. The rotate range of each joint is [ ?60?,+60?]. The rotating shaft is 45#steel. The shell of the snake-like robot is printed by a 3D printer using high-strength nylons.

To test robot’s performance of resisting external collision,experiments similar with above simulations are conducted as shown in Fig. 11. The mass of the impactor is about 1.5 kg.The initial state of the robot is a straight line, joints’ angles are set as ?i=0.

Similarly to Fig. 9, two groups of experiments are made of aluminum alloy and other parts of MFB are made of conducted in ADAMS as shown in Fig. 11. In the first group,the robot’s stiffness keep constant as shown in Fig. 11(a). In the second one, the robot adjusts its stiffness by using methods proposed in Section III as shown in Fig. 11(b). It is seen that the robot with adaptive controlled stiffness is stabler than the robot with constant stiffness when they are stricken by the same impactor.

Experiments in Fig. 12 are designed to verify the simulations in Fig. 9. Since the motors are driven by pulses,we solve motors’ pulse frequencies by using joint angles (as shown in Fig. 7) from the controller. As shown in Fig. 12, the stiffness of the robot is constant in Fig. 12(a) and is increased in Fig. 12(b). From results in Fig. 12, it is concluded that the displacement of a robot with variable stiffness (Fig. 12(b)) is less than the displacement of another robot with constant stiffness (Fig. 12(a)).

Experiment results show that the adaptive controlled stiffness is useful to improve the snake-like robot’s stability.

Fig. 7. Displacement and joint angles of the snake-like robot. (a) Displacement. (b) Joint angles.

Fig. 8. The solved stiffness values in (a) the x direction and (b) the y direction. Torques of (c) the first joint and (d) the fourth joint.

TABLE II Parameters of the Snake-like Robot

Fig. 9. Snake robots are hit by impactors in ADAMS.

V. Conclusion

In this work, a snake-like robot with variable stiffness actuators is analysed. An adaptive stiffness control method based on stiffness-energy optimization is proposed for the robot.A velocity decomposition and an exponential stabilization controller are used to construct the dynamic and motion control of the robot. To control stiffness and optimize energy consumption, a multi-objective evolutionary algorithm is researched. Simulations and experiments are conducted to verify the proposed adaptive stiffness control and optimization methods for the snake-like robot with variable stiffness actuators.

Fig. 10. Displacements of snake robots in the y direction when they are hit by impactors.

Fig. 11. Effect of external impact on the shape of a snake-like robot with(a) unchanged stiffness and (b) increased stiffness.

Fig. 12. Experiment of variable stiffness performance for maintaining the posture robot with (a) unchanged stiffness and (b) increased stiffness.