Bipedal Locomotion Control via CPGs with - IEEE Xplore

Proceedings of International Conference on Mechatronics Kumamoto Japan, 8-10 May 2007

TuA2-B-5

Bipedal Locomotion Control via CPGs with Coupled Nonlinear Oscillators Maki K. Habib, Guang Lei Liu, Keigo Watanabe and Kiyotaka Izumi Department of Advanced Systems Control Engineering Graduate School of Science and Engineering Saga University, Japan {habib, watanabe, izumi} @me.saga-u.ac.jp, liuguanglei @hotmail. com a motion trajectory that guarantees a dynamic stability [1]-[3]. A walking pattern or reference trajectory is typically computed offline so as to satisfy the necessary ZMP constraints and online tracking is then used to realize the desired trajectory at any instance by ensuring surface contact between the sole and the ground. In general, the ZMP based control algorithms for locomotion have been shown to be effective for controlling the posture and low-speed walking of biped and quadruped robots with flat feet under controlled environments. However, they require pre-recorded trajectories with precise modeling of both the robot and the environment, and precise joint actuation with high control gains. In addition, ZMP concept is not suitable for medium- or high-speed walking from the standpoint of energy consumption, because a body with a large mass needs to be accelerated and decelerated by actuators in every step cycle. More significantly, in order to avoid inverse-kinematic singularities, their robots result in 'bent-knee' motions that do not look natural, which cause significant limits on the types of walking styles that should be generated [4]. Another control scheme called virtual model control [5], [6] has been developed. This technique is a motion control language which uses simulations to introduce a number of virtual components and construct a controller. Due to the intuitive nature of this technique, designing a virtual model controller requires the same skills as need for designing the mechanism itself, because it involves many manually-tuned control parameters and does not allow for the direct specification of a desired motion style. However, from the biological point of view, locomotion of humans and animals does not require such precision and compensation, and it is quite different from that of current biped robots. There are evidences showing the existence of various oscillatory or rhythmic pattern generation activities within the neural circuitry in almost every animal, and most of them are produced without receiving any particular extrinsic oscillatory stimulus [7]. It has become generally accepted that animals' walking involves rhythmic activities that are mainly generated at the spinal cord by a combination of CPGs and reflexes receiving adjustment signals from a cerebrum, cerebellum and brain stem [8], [9], [14]. Almost all species developed completely different form of locomotion perfectly suited to its morphology and environment to ensure its sur-

Abstract-This paper describes the design and analysis of a biologically inspired central pattern generator (CPG) using a network of mutually coupled nonlinear oscillators to generate rhythmic walking pattern for biped robots. The paper examines the characteristics of a CPG model composed of a network of nonlinear oscillators, and the effect of assigning symmetrical and asymmetrical coupling mechanism among oscillators within the network structure under different possibilities of inhibitions and excitations. The paper highlights the necessity to understand the targeted physical system and its functionalities before concluding the design parameters of the CPG. In addition, the paper considers the way in which the sensory feedback contributes to generate adaptive walking trajectory and enhance gait stability, and how the driving input and external perturbation affect the speed of locomotion and change the period of its own active phase. Modeling of bipedal robot using a CPG based controller and a musculo-skeletal system has been achieved for the purpose to realize the interaction with each other and to study the necessary conditions for stable dynamic walking on dynamic terrain that lead to stable and sustained response from the network. The kinematics and dynamics of a five-link biped robot has been modeled and its joints are actuated through simulation by the proportional torques output from the CPG to generate the trajectories for hip, knee, and ankle joints. The CPG based bipedal locomotion is carried out and evaluated through simulations using MATLAB. I.

INTRODUCTION

Walking bipeds are high-dimensional nonlinear systems with complexity of large degrees of freedom, and there is still lack of powerful techniques for its control. The dynamic legged locomotive robots are challenging to control because they are operated throughout the range of their state space, act in a gravity filed, interact with dynamic environment, exhibit time variant dynamics, etc. These robots require developing new technologies and creating powerful techniques to overcome wide range of issues, such as motion control, torque control, stability, trajectory generation, generation of different gait patterns depending on the state of the environment, sensing and perception, etc. Locomotion is one of the critical issues that need to be addressed properly and efficiently. The development in this field has enabled humanoid robots to navigate real environments [1]. Key elements in such development were the ZMP (Zero Moment Point) based control approaches, which aim at compensating the rolling and pitching moment of the robots. ZMP based methods are used to define 1-4244-1184-X/07/$25.00©2007 IEEE

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vival. The fundamental mechanisms of animals' rhythmic biological movement, such as locomotive motion of biped and quadruped, flapping of bird wings, swimming of fish, crabs, respiration, heartbeat, etc. are typical examples of oscillatory activities that have been studied both in biological science and in engineering. Studies of locomotory control systems in living organisms have led to the formulation of a spinal circuit known as neuron oscillators or CPGs [10]. A CPG generates coordinated multidimensional rhythmic motor patterned outputs under the control of simple input signals. This occurs because most motor activities in the animals are performed by reflexes and not by cerebral activities. The CPGs are found in both vertebrate and invertebrate animals for the control of their locomotive activities. However, human locomotion is partly controlled by a CPG [17]. The interest in CPG is not only to observe the behavior of oscillatory neural network, but also to find the mechanisms through which oscillations take place [11]. It has been suggested that sensory feedback plays an important role in stabilizing rhythmic movements by coordinating the high level activity (nervous system), physical system and the CPGs [13]. But, how the sensory feedback can be integrated into the CPG and interacts with it to enable the adaptation with the environment and increase the stability of the gait is still an open area of research [12]. Biologically inspired locomotion control approaches based on CPGs have drawn much attention from many researchers to generate rhythmic motion for biped robots that resemble human-like gait featured with self-adaptive properties to cope with changes in their environment [13], [15], [16]. CPG can be represented by different mathematical models such as oscillators, artificial neurons, vector fields, etc. The use of a set of mutually coupled nonlinear oscillators is widely used approach in which each oscillator has its proper amplitude, frequency, coupling terms, and other parameters [7], [11], [12], [15], [18]-[21]. Matsuoka [7], [18] proposed a mathematical model and analyzed the conditions for mutual inhibitory networks represented by a continuous-variable neuron model to generate any oscillation. Matsuoka model is widely used to model the firing rate of two mutually inhibitory neurons described in a set of differential equations. This model is used in robotic applications to achieve designated tasks involving rhythmic motion which requires interactions between the system and the environment. However, it is very difficult to set the CPG parameters for various robots and environments, because there is no design principle yet to determine the proper settings. Taga et al. proposed the principle of adaptive control of locomotion system, in which the nervous, musculo-skeletal, and sensory systems behave cooperatively and adapt to unpredictable environments [12]. The result of biped walk through simulations has been demonstrated through several researches. In addition, the motion control method using nonlinear oscillators has been proposed for stable dynamic walking of quadruped robots [20] and a hexapod robot [21]. A system of coupled adaptive nonlinear oscillators, which can learn arbitrary rhythmic signals in a supervised learning framework, has been used as

programmable CPGs to control the locomotion of a humanoid robot [19]. In spite of all the progress in the development of CPG based techniques, a key drawback that affects its progress is the need to tune the design parameters in order to match it for a specific application. Currently, it is difficult to find a systematic technique that adapts for a CPG aiming to generate an arbitrary periodic signal. This paper introduces the design and analysis of a biologically inspired central pattern generator (CPG) using a network of mutually coupled nonlinear oscillators to generate rhythmic walking pattern for biped robots. The paper highlights the necessity to understand the targeted physical system and its functionalities before concluding the design parameters of the CPG. The CPG-based biped walk is carried out and evaluated through simulations. II. THE CENTRAL PATTERN GENERATORS

CPG is a model of biologically inspired rhythmic system, and consists of nonlinear neural oscillators where the mutual inhibition among neurons is modeled such that neuron's excitation suppresses others neurons' excitations. CPGs encode rhythmic trajectories as limit cycles based control behavior of coupled nonlinear dynamical systems. Stable limit cycle behaviors are featured with properties that will return the dynamic system to stable state after a certain range of perturbations, and they are almost not influenced by a change in the initial conditions. It has been pointed out that a CPG has the capability to generate and modulate walking patterns and to be mutually entrained with a rhythmic joint motion [8], [9], [12]. In addition, the motion generated has better energy efficiency [20]. The neural system model consists of a CPG and reflexes, and there are two types of implementations for such combination: 1) CPG produces desired positions, and 2) CPG produces output proportional to torques. Different models could be used to represent the interaction between the CPG and the reflex system that represents the type of the feedback mechanism from internal sensory information and the interaction with robot environment. In order to represent a CPG and generate the required signals several nonlinear oscillators that are coupled together have been developed, such as, Hopf, Rayleigh, Van del Pol, Matsuoka, etc. oscillators. In this paper, Matsuoka nonlinear neural oscillator with two reciprocally linked and mutually inhibited neurons was used as a basic building block, and accordingly the mathematical model of the oscillator was introduced into the locomotion of bipedal robot. The CPG model and the reflexes in this research produce outputs proportional to the required torque at each joint. The model of each neuron has two state variables and it is represented by two equations as below:

dui

TIr dt

dfi

2

n

-U

+

I WjYj + WsiSo

bfi + feedi (1)

j=l

Tai dt

-fi + Yi

(2)

Yi (ui)

max{0, ui}

(3)

IExternal events High level activfty coordinator to activate the relevant neural rhythm

generators

I

,

~~~~~Network of coupled neural

l Feedback signals

Ground contact forces Joint positions and velocities - Internal sensory infonnation -

(a)

-

tor

parameters |~~~~~~CPG oscillators to generate proper

rhythmic signals

n Musculo-skeletal system Dynamic equation of motion using

Newton-Euler method

|Actual trajectories

Fig. 2. The general control strategy for the bipedal robot.

Yj

inhibition is done through wijyj and wjiyi. The output from the oscillator is Tor = ajyj -ai yi that represents the algebraic sum of the weighted output signal from each neuron, where ai and aj are constant gains. The To, can be used as a motor command to drive a joint of one DOF, where To, > 0 implies the dominant activity of extensor neuron, and To, < 0 means the dominant activity of flexor neuron. Fig. l(b) shows two coupled neurons formulating Matsuoka oscillator.

Yi

(b) Fig. 1. (a) General Matsuoka neuron model; (b) A nonlinear oscillator consisting of an extensor and a flexor.

III. CPG AND BIPED ROBOT FOR STABLE DYNAMIC

RHYTHMIC GAIT The first state variable is ui that corresponds to the membrane potential of the jth neuron body, and the second state variable A. Framework of Control Strategy for Gait Generation is fi that represents the degree of fatigue or the degree of The transformation from high level task specification to low adaptation (self-inhibition) in the neuron, while yi represents level motion control is a fundamental issue in sensorimotor the positive part of ui. The subscripts i, j denote the neuron control in animals and robots. Accordingly, Fig. 2 introduces number, Ti is the time constant that specifies the rise time the adopted framework of the control strategy for the bipedal when given a step input. The frequency of the output is robot aiming to generate flexible rhythmic walking gait, and roughly proportional to 1/Ti. In addition, Tai is the time it includes three parts. The first part consists of two major constant that specifies the time lag of the adaptation effect. elements. The first element represents the high level activity wij denotes inhibitory synaptic connection weight from the coordinator that can set and activate the relevant neural rhythjth neuron to the ith neuron; when i j, the value of the mic motion based on external and internal sensory information. connection weight can be wij < 0 and wij = 0, when i j. The second element within the first part represents the network In addition, E wijyj represents the total input from other of coupled neural oscillators aiming to generate synchroneurons within the network of neural oscillators. The so is the nized rhythmic signals. The locomotor movement results from external high level driving input with constant rate, and wso torques generated by the neural rhythm generator and acting at denotes a connection weight of the driving input, and feedi each joint of the robot. The second part of the control strategy is feedback sensory signal to the jth neuron and represents includes the model of the musculo-skeletal system along with the internal sensory information and interaction between the the mathematical formulation of the dynamic equations of robot and its environment. The mathematical conditions to motion using Newton-Euler method. The last part represents generate oscillations are analyzed in [7], [18]. The feedi the feedback signals that aim to establish a closed-loop to has been added to the neuron model of Matsuoka to build enable real time adaptation for the walking gait. a close-loop dynamic system model with feedback sensory B. Biped Musculo-Skeletal Model information [12]. 1 shows the Matsuoka neuron model deFig. (a) general Fig. 3 shows the simple model for a bipedal musculoscribed by Eqs. (1), (2) and (3). The activity of the neural skeletal system that has been considered in this paper. It oscillator is used to account for the alternating activities of has six joints and two identical legs each with three DOFs flexor and extensor at each joint during walking. The self- corresponding to hip, knee and ankle joints. Each leg is inhibition is governed by the bfi connections while the mutual composed of a thigh (links 2 and 3), and a shank (links 4

3

Link 1

Link 3

2 Er t~or2

F K Tor4I

y

i

FE

Or

Link 2

Knee joints E

F

N) E Tor3

Link 5 tor6

Reference x frame

T-

The constraint forces can be obtained by substituting Eq. (4) into Eq. (5), and to get the required accelerations without using the constraint forces, the yielded equation of forces is substituted into Eq. (4). Hence, the compact form of the acceleration equations that represents the motion of the bipedal musculo-skeletal system is,

F Hip joints

z

Link 4

F

Ankle joints

E

F

v

Left leg

Right leg

and 5). In the dynamic model of the bipedal musculo-skeletal system, the links are considered to be of uniform rectangular shape with mass at its center. A point mass is used to represent the remaining part of the body and it is described by link 1 at the hip. Both legs are integrated at link 1 while assuring suitable detachment. The joints are numbered as Joints 1, 2, 3, 4, 5 and 6 from the side of the body, where Joints 1 and 2 are the hip joints, Joints 3 and 4 are knee joints, and Joints 5 and 6 are ankle joints. Due to their low inertia, a point foot has been considered into the dynamics during the support phase, and contact with the ground has been represented by two dimensional spring and damper. Vertical and horizontal ground reaction forces are modeled and calculated during the time that ankles make contact with the ground respectively. A slippage model is established by using a condition that manages the relation between both reactive forces and the static friction coefficient of the ground. In addition, the described model is bounded to move within the sagittal plane, and the torques acting at the joints to realize the walking gait are assumed to be generated by the CPG controller.

P(x)F + Q(x,

,

Tr(y), Fg (xJ))

(6)

D. The Model for the CPG Based Controller The nonlinear neural oscillators are the main elements that compose the model of the CPG. The simplest model of the neural oscillator consists of two mutually inhibited neurons with self adaptation in each of them and this constitutes a CPG unit. Each CPG unit is dedicated to control one joint, and it has four state variables. Two variables represent the inner states of each neuron (ui and uj), and the other two state variables represent the degree of adaptation for each neuron (fi and fj), respectively. Six mutually coupled CPG units were used to model the CPG based controller for the bipedal robot. One CPG unit was used at each side of the hip (left and right) and one unit was used at each of the knee and ankle joints and at both sides of the robot model. By connecting the CPG units of each leg at the hip level, the CPGs are mutually entrained and oscillate in the same period and with a fixed phase difference. Hence, the mutual entrainment between the CPGs of the legs results in a gait. Fig. 4 illustrates the arrangement of the neural oscillators in relation to the adopted bipedal musculoskeletal system. The neurons with odd number represent flexor the (F) and those with even number represent the extensor (E), while Tori to Tor6 represent the output torques from the oscillators. The configured neural oscillators have inhibitory connections. The two neurons of each CPG unit generate torques in opposite direction, i.e., the direction of contraction of flexor and extensor muscles. The algebraic sum of the torques at each neural oscillator is proportional to the torque at the relevant joint during biped walk. The inhibitory connection between the hip CPG units produces alternate excitations with half cycle difference to facilitate the alternation between the movements of the two legs. In addition, all the CPG units have stable limit cycles that are largely determined by the dynamics of the CPG network, whose dynamic characteristics can be adjusted during operation by the external driving inputs. Here, their inputs consist of the sensory feedback from legs and the

C. The Mathematical Motion Formulation of the Model By using the Newton-Euler dynamic formulation, the general form of equations of motion for the bipedal musculoskeletal system is derived as below [12]: =

P (x) [C(x)P(x) -l[D(x, --) C (x) Q (x,.-., T, (y), Fg (x, --.))] +Q (x .-. . Tr (y) . Fg (x

To solve the motion equations, the y vector is provided as an output from the CPG based controller that is proportional to the torque required at each joint. In addition, the feedback signal from the bipedal robot to the neural rhythm generator is represented by the joints' inertial positions and velocities of different moving parts of the robot's body, and the contact forces with the environment. In case of a physical biped robot, the feedback sensory information is sensed through different internal and external sensors.

Fig. 3. Five link model for a biped locomotion.

Xc

=

(4)

where x C R14 is a vector of the inertial positions of 5 links and the inertial angles of 4 links; p C R14x8 is a matrix; F C R8 is a vector of constraint forces; Q C R14 is a vector; Tr C ER6 is a vector of torques; Fg C R4 is a vector of forces on the ankle which depends on the state of the terrain; and y C 12 is a vector of the neural rhythm generator output. From the biped model, the equations of the kinematic constraints are formulated, and hence the acceleration can be obtained by differentiating these equations twice with respect to time. The yielded equations can be written in the following compact form, C (x)j = D (x, t)(5) 4

F 1.5

E

E. I1 a)

° 05

t