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BIPED HOPPING CONTROL BASED ON SPRING LOADED INVERTED PENDULUM MODEL

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Human running can be stabilized in a wide range of speeds by automatically adjusting muscular properties of leg and torso. It is known that fast locomotion dynamics can be approximated by a spring loaded inverted pendulum (SLIP) system, in which leg
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  1 st   Reading May 19, 2010 17:41 WSPC/S0219-8436 191-IJHR 00210 International Journal of Humanoid Robotics 1 Vol. 7, No. 2 (2010) 1–18c  World Scientific Publishing Company 3 DOI:10.1142/S0219843610002106 5 BIPED HOPPING CONTROL BASED ON SPRING LOADEDINVERTED PENDULUM MODEL 7 SEYED HOSSEIN TAMADDONI ∗ , FARID JAFARI † ,ALI MEGHDARI ‡ and SAEED SOHRABPOUR § 9 Center of Excellence in Design,Robotics, and Automation (CEDRA), 11 School of Mechanical Engineering,Sharif University of Technology, Tehran, Iran  13 ∗ h.tamaddoni@gmail.com  †  jafari.farid@gmail.com  15 ‡ meghdari@sharif.edu  § sohrabpour@sharif.edu  17 ReceivedRevised 19 AcceptedHuman running can be stabilized in a wide range of speeds by automatically adjusting 21 muscular properties of leg and torso. It is known that fast locomotion dynamics canbe approximated by a spring loaded inverted pendulum (SLIP) system, in which leg is 23 replaced by a single spring connecting body mass to ground. Taking advantage of theinherent stability of SLIP model, a hybrid control strategy is developed that guarantees 25 a stable biped locomotion in sagittal plane. In the presented approach, nonlinear controlmethods are applied to synchronize the biped dynamics and the spring-mass dynamics. 27 As the biped center of mass follows the mass of the mass-spring model, the whole bipedperforms a stable locomotion corresponding to SLIP model. Simulations are done to 29 obtain a repeatable hopping for a three-link underactuated biped model. Results showthat periodic hopping gaits can be stabilized, and the presented control strategy provides 31 feasible gait trajectories for stance and swing phases. Keywords : Biped; hopping; jumping; running; SLIP; gait; control. 33 1. Introduction For decades, motion planning has been a crucial step in developing walking robots. 35 In particular, there is significant interest on developing systematic methods of gaitgeneration for biped robots. 1 – 4 Among the existing methods, the objective function 37 oriented approach that regulates the motion of a planar link-segmental biped robotsuch that the cost is minimized has shown considerable advantages. Through this 39 method, biped locomotion can be characterized in terms of step length, progressionspeed, maximum step height and the stance knee bias angle. According to the objec- 41 tive function, joint angular displacement profiles can be then uniquely determinedbased on the initial condition of the system. 43 1  1 st   Reading May 19, 2010 17:41 WSPC/S0219-8436 191-IJHR 00210 2 S. H. Tamaddoni et al. Despite different gaits and speeds, human and legged animal fast locomotion 1 can be described by spring-like leg behavior 5 ; hence, fast locomotion including run-ning, jumping, hopping, and etc. can be represented by a simple spring-mass sys- 3 tem, known as Spring Loaded Inverted Pendulum  or SLIP. 6 , 7 This model is basedon observations which revealed that the energy level remains approximately con- 5 stant during running, hopping and jumping. Associated with the spring-like legbehavior, leg stiffness is defined that is shown to remain almost constant during 7 fast locomotion. 8 Thus, the spring-mass representation seems to provide acceptableinsight to human fast locomotion. 9 , 10 It is also known that the spring-mass model 9 is repeatable and self-stabilizing for adequate leg parameter adjustments, includingangle of attack, leg stiffness, and leg length, and sufficient speeds. 8 , 11 11 In this paper, a new approach to biped trajectory planning and control in hop-ping locomotion is introduced based on synchronization of biped governing dynam- 13 ics and SLIP dynamics that corresponds to a particular hopping motion. Thismethod is biologically inspired by the idea that biped must follow human trajectory 15 which is considered an ideal self-stabilizing pattern in order to become capable of fast locomotion. 12 , 13 17 A link-segmental model of the biped with three degrees of freedom and onedegree of underactuation in the ankle joint is developed in this paper. Using the 19 proposed control strategy, the biped joint torques are commanded such that thebiped dynamics follows its corresponding SLIP model. Later discussed, the initial 21 conditions are set in a way that a periodic hopping motion is achieved. Simulationresults show a periodic hopping locomotion can be achieved using the proposed 23 synchronization control method. 2. Dynamics Model of Hopping 25 The robot is assumed to consist of three rigid links with mass, connected via rigid,frictionless, revolute joints to for a single open kinematic chain lying in sagittal 27 plane. The hopping occurs on a flat horizontalsurface, thus without loss of generalityit is assumed that the ground height is zero with respect to the inertial frame. 29 A complete cycle of hopping consists of two consecutive phases:(1) stance in which leg end is in stationary contact with the ground and as an ideal 31 pivot supports the whole body. If the biped center of mass (CoM) reaches asufficient upward velocity, the leg loses its contact to the ground and the phase 33 ends,(2) flight  in which there is no contact with the ground, and it ends as soon as a 35 sudden impact with the ground surface occurs.Figure 2 illustrates different phases of running with coordinate conventions of this 37 paper labeled on a link-segmented model of biped robot in sagittal plane. The bipedmodel in this study has three links each of which has length, mass, and moment 39  1 st   Reading May 19, 2010 17:41 WSPC/S0219-8436 191-IJHR 00210 Biped Hopping Control Based on Spring Loaded Inverted Pendulum Model  3 of inertia. There is one degree of underactuation in ankle joint of the biped, so the 1 robot cannot apply any torque at its foothold.Assuming that θ, ˙ θ, ¨ θ , and τ  represent the generalized coordinates, velocities,accelerations, and torques, respectively, the dynamic model describing the motionof the biped in stance phase can be written as, 14 D ( θ )¨ θ + C  ( θ, ˙ θ )˙ θ + G ( θ ) = Bτ  (1)where D ( θ ) is the inertia matrix, C  ( θ, ˙ θ ) is the Coriolis matrix, G ( θ ) is the gravity 3 vector, and B is the control matrix.For i,j = 1 , 2 , 3, the matrices D,C,G are given by:  D ij ( θ ) =  m j d j l j +  3  k = j +1 m k  l i l j  cos( θ i − θ j ) C  ij ( θ, · θ ) =  m j d j l j +  3  k = j +1 m k  l i l j  sin( θ i − θ j ) · θ j G i ( θ ) =  m j d j g +  3  k = i +1 m k  l i g  sin θ i . (2)Since the biped is underactuated in the contact point, no torque can be exertedfrom the ground to the first link. Therefore, the control matrix B corresponding tothe generalized torque vector τ  = [ τ  2 τ  3 ] T  is given by, B =  − 1 01 − 10 1  (3)During flight phase, five degrees of freedom including joint angles and ankle position 5 are defined to describe the governing dynamics. The biped center of mass undergoesa ballistic trajectory, and the exerted torques in knee and hip defines the angle 7 trajectory during flight. Given that the angle trajectories of ankle and knee jointsare known, the hip angle and ankle position is determinant by flight dynamics. 9 When the ankle touches the ground at the end of flight phase, an impact takesplace. Based on the existing impact models, the ground reaction forces at impact can 11 be represented by impulses with intensity I  R . In this paper, the impact is assumedinelastic, and the velocity of the ankle becomes zero instantaneously. As a result 13 of this model, the robot’s configuration is assumed unchanged during impact andthere are only instantaneous changes in the velocities. 15 The Cartesian position of the ankle can be expressed in terms of Cartesianposition of biped center of mass as,  x a y a  =  x c y c  − s ( θ ) (4)  1 st   Reading May 19, 2010 17:41 WSPC/S0219-8436 191-IJHR 00210 4 S. H. Tamaddoni et al. where s is determined from the in-between link parameters such as length, mass, 1 and position of center of mass.Let˙ θ − , ˙ θ + denote the velocity vector just before and after the impact, respec-tively. After impact, the ankle neither rebounds nor slides; therefore, the linearvelocity of the center of mass can be expressed in terms of the angular velocities just after impact as,  00  =  ˙ x + c ˙ y + c  − ∂s ( θ ) ∂θ ˙ θ + . (5)The ground reaction impulse vector is given by, I  R = m  ∂s ( θ ) ∂θ ˙ θ + −  ˙ x − c ˙ y − c  (6)and the robot’s angular velocity vector after impact is obtained with respect to thevelocity before impact as,˙ θ + =  A (˜ θ ) + m∂s T  ( θ ) ∂θ∂s ( θ ) ∂θ  − 1  A (˜ θ ) m∂s T  ( θ ) ∂θ  ˙ θ − (7)where˜ θ is the relative angles of the actuated joints with reference to the bipedbody, and A is the matrix that defines the total kinetic energy of the robot duringflight, K  f  =12˙˜ θ T  A ˙˜ θ +12 m (˙ x 2 c + ˙ y 2 c ) . (8) 3. Spring Loaded Inverted Pendulum Model 3 The planar spring model is characterized by alternating stance and flight phases.Figure 1 illustrates the parameters of the spring mass model. The point mass m 5 Fig. 1. Three-link model of biped robot.  1 st   Reading May 19, 2010 17:41 WSPC/S0219-8436 191-IJHR 00210 Biped Hopping Control Based on Spring Loaded Inverted Pendulum Model  5Fig. 2. Spring mass running model. represents the effective body mass, which is supported by a linear spring of stiffness 1 k and rest length l 0 . During stance phase, the spring is pinned to the ground withthe angle of attack α between the neutral axis and the point mass. During flight 3 phase, the center of mass describes a ballistic curve, determined by the gravitationalforce acting downward. 5 The transition to stance phase occurs when the following landing condition issatisfied: y = l 0 sin( α TD ) (9)where α TD is the touch-down angle of attack.During stance phase, spring mass dynamics is governed by spring forces andgravity, and the equation of motion is:  m ¨ l = ml ˙ α 2 − mg cos α − k ( l − l 0 ) ml ¨ α = − 2 m ˙ l ˙ α + mg sin α (10)where l,l 0 are the spring current length and rest length, respectively, and α is the 7 angle of attack.Assuming small angle of attack, α , an approximated analytical solution can be 9 obtained for the spring mass system of Eq. (2). 9 However, since biped hops in awide range of angle of attacks, the equations of motion for spring mass system is 11 solved numerically in this work.During flight phase, the center of mass describes a ballistic curve, determinedby the gravitational force acting downward:  ¨ x c = 0¨ y c = − g. (11)Combining equations (10) and (11), a hybrid model is formed that describes the 13 dynamics of spring mass in stance and flight.Since, the horizontal component of center of mass velocity, v x , remains 15 unchanged during flight phase, its value should be the same at touchdown andtakeoff to guarantee periodicity of hopping motion. This implies that symmetry of  17
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