Walking C H A P T E R 5

Transcription

1 C H A P T E R 5 Waking q Practica egged ocomotion is one of the fundamenta unsoved probems in robotics. Many chaenges are in mechanica design - a waking robot must carry a of it s actuators and power, making it difficut to carry idea force/torque - controed actuators. But many of the unsoved probems are because waking robots are underactuated contro systems. In this chapter we introduce some of the simpe modes of waking robots, the contro probems that resut, and a very brief summary of some of the contro soutions described in the iterature. Compared to the robots that we have studied so far, our investigations of egged ocomotion wi require additiona toos for thinking about imit cyce dynamics and deaing with impacts. 5. LIMIT CYCLES A imit cyce is an asymptoticay stabe or unstabe periodic orbit. One of the simpest modes of imit cyce behavior is the Van der Po osciator. Let s examine that first... EXAMPLE 5. Van der Po Osciator q + µ(q ) q + q = One can think of this system as amost a simpe spring-mass-damper system, except that it has noninear damping. In particuar, the veocity term dissipates energy when q >, and adds energy when q <. Therefore, it is not terriby surprising to see that the system settes into a stabe osciation from amost any initia conditions (the exception is the state q =, q = ). This can be seem nicey in the phase portrait in Figure 5.(eft) qdot q t FIGURE 5. System trajectories of the Van der Po osciator with µ =.. (Left) phase portrait. (Right) time domain. marginay-stabe orbits, such as the cosed-orbits of the undamped simpe penduum, are typicay not caed imit cyces. 4 c Russ Tedrake, 9

2 Section 5. Poincaré Maps 43 However, if we pot system trajectories in the time domain, then a sighty different picture emerges (see Figure 5.(right)). Athough the phase portrait ceary reveas that a trajectories converge to the same orbit, the time domain pot reveas that these trajectories do not necessariy synchronize in time. The Van der Po osciator ceary demonstrates what we woud think of as a stabe imit cyce, but aso exposes the subtety in defining this imit cyce stabiity. Neighboring trajectories do not necessariy converge on a stabe imit cyce. In contrast, defining the stabiity of a particuar trajectory (parameterized by time) is reativey easy. Let s make a few quick points about the existence of cosed-orbits. If we can define a cosed region of phase space which does not contain any fixed points, then it must contain a cosed-orbit[83]. By cosed, I mean that any trajectory which enters the region wi stay in the region (this is the Poincare-Bendixson Theorem). It s aso interesting to note that gradient potentia fieds (e.g. Lyapunov functions) cannot have a cosed-orbit[83], and consquenty Lyapunov anaysis cannot be appied to imit cyce stabiity without some modification. 5. POINCAR É MAPS One definition for the stabiity of a imit cyce uses the method of Poincaré. Let s consider an n dimensiona dynamica system, ẋ = f(x). Define an n dimensiona surface of section, S. We wi aso require that S is tranverse to the fow (i.e., a trajectories starting on S fow through S, not parae to it). The Poincaré map (or return map) is a mapping from S to itsef: x p [n + ] = P(x p [n]), where x p [n] is the state of the system at the nth crossing of the surface of section. Note that we wi use the notation x p to distinguish the state of the discrete-time system from the continuous time system; they are reated by x p [n] = x(t c [n]), where t c [n] is the time of the nth crossing of S. EXAMPLE 5. Return map for the Van der Po Osciator Since the fu system is two dimensiona, the return map dynamics are one dimensiona. One dimensiona maps, ike one dimensiona fows, are amenabe to graphica anaysis. To define a Poincare section for the Van der Po osciator, et S be the ine segment where q =, q >. If P(x p ) exists for a x p, then this method turns the stabiity anaysis for a imit cyce into the stabiity anaysis of a fixed point on a discrete map. In practice it is often difficut or impossibe to find P anayticay, but it can be obtained quite reasonaby numericay. Once P is obtained, we can infer oca imit cyce stabiity with an eigenvaue anaysis. There wi aways be a singe eigenvaue of - corresponding to perturbations aong the imit cyce which do not change the state of first return. The imit cyce is considered ocay exponentiay stabe if a remaining eigenvaues, λ i, have magnitude ess than one, λ i <. In fact, it is often possibe to infer more goba stabiity properties of the return map by examining, P. [44] describes some of the stabiity properties known for unimoda maps. c Russ Tedrake, 9

3 44 Chapter 5 Waking qdot qdot p [k+] q qdot p [k] FIGURE 5. (Left) Phase pot with the surface of section, S drawn with a back dashed ine. (Right) The resuting Poincare first-return map (bue), and the ine of sope one (red). A particuary graphica method for understanding the dynamics of a onedimensiona iterated map is with the staircase method. Sketch the Poincare map and aso the ine of sope one. Fixed points are the crossings with the unity ine. Asymptoticay stabe if λ <. Unike one dimensiona fows, one dimensiona maps can have osciations (happens whenever λ < ). [insert staircase diagram of van der Po osciator return map here] 5.3 THE BALLISTIC WALKER One of the eariest modes of waking was proposed by McMahon[59], who argued that humans use a mosty baistic (passive) gait. COM trajectory ooks ike a penduum (roughy waking by vauting). EMG activity in stance egs is high, but EMG in swing eg is very ow, except for very beginning and end of swing. Proposed a three-ink baistic waker mode, which modes a singe swing phase (but not transitions to the next swing nor stabiity). Interestingy, in recent years the fied has deveoped a consideraby deeper appreciation for the roe of compiance during waking; simpe waking-by-vauting modes are starting to fa out of favor. McGeer[55] foowed up with a series of waking machines, caed passive dynamic wakers. The waking machine by Coins and Ruina[6] is the most impressive passive waker to date. 5.4 THE RIMLESS WHEEL The most eementary mode of passive dynamic waking, first used in the context of waking by [55], is the rimess whee. This simpified system has rigid egs and ony a point mass at the hip as iustrated in Figure 5.3. To further simpify the anaysis, we make the foowing modeing assumptions: Coisions with ground are ineastic and impusive (ony anguar momentum is conserved around the point of coision). The stance foot acts as a pin joint and does not sip. The transfer of support at the time of contact is instantaneous (no doube support phase). c Russ Tedrake, 9

4 Section 5.4 The Rimess Whee 45 m α γ θ g FIGURE 5.3 The rimess whee. The orientation of the stance eg, θ, is measured cockwise from the vertica axis. γ < π, < α < π, >. Note that the coordinate system used here is sighty different than for the simpe penduum (θ = is at the top, and the sign of θ has changed). The most comprehensive anaysis of the rimess whee was done by [4] Stance Dynamics The dynamics of the system when one eg is on the ground are given by θ = g sin(θ). If we assume that the system is started in a configuration directy after a transfer of support (θ( + ) = γ α), then forward waking occurs when the system has an initia veocity, θ ( + ) > ω, where ω = g [ cos (γ α)]. ω is the threshod at which the system has enough kinetic energy to vaut the mass over the stance eg and take a step. This threshod is zero for γ = α and does not exist for γ > α. The next foot touches down when θ(t) = γ + α, at which point the conversion of potentia energy into kinetic energy yieds the veocity g θ (t ) = θ ( + ) + 4 sin α sin γ. t denotes the time immediatey before the coision Foot Coision The anguar momentum around the point of coision at time t just before the next foot coides with the ground is c Russ Tedrake, 9 L(t ) = m θ (t ) cos(α).

5 46 Chapter 5 Waking The anguar momentum at the same point immediatey after the coision is L(t + ) = m θ (t + ). Assuming anguar momentum is conserved, this coision causes an instantaneous oss of veocity: θ (t + ) = θ (t ) cos(α). The deterministic dynamics of the rimess whee produce a stabe imit cyce soution with a continuous phase punctuated by a discrete coision, as shown in Figure 5.4. The red dot on this graph represents the initia conditions, and this imit cyce actuay moves counter-cockwise in phase space because for this tria the veocities were aways negative. The coision represents as instantaneous change of veocity, and a transfer of the coordinate system to the new point of contact. dθ/dt dθ/dt θ θ dθ/dt dθ/dt θ θ FIGURE 5.4 Phase portrait trajectories of the rimess whee (m =, =, g = 9.8, α = π/8, γ =.8) Return Map We can now derive the anguar veocity at the beginning of each stance phase as a function of the anguar veocity of the previous stance phase. First, we wi hande the case where γ α and θ + > ω. The step-to-step return map, factoring osses from a singe n c Russ Tedrake, 9

6 Section 5.4 The Rimess Whee 47 coision, the resuting map is: θ + = cos(α) (θ n + ) + 4 g n+ sin α sin γ. where the θ + indicates the veocity just after the energy oss at impact has occurred. Using the same anaysis for the remaining cases, we can compete the return map. The threshod for taking a step in the opposite direction is For ω < θ + < ω, we have n ω = g [ cos(α + γ)]. θ + = θ + cos(α). n+ n Finay, for θ + n < ω, we have g θ + n+ = cos(α) (θ + n ) 4 sin α sin γ. Notice that the return map is undefined for θ n = {ω, ω }, because from these configurations, the whee wi end up in the (unstabe) equiibrium point where θ = and θ =, and wi therefore never return to the map. This return map bends smoothy into the case where γ > α. In this regime, cos(α) (θ n + ) + 4 g sin α sin γ, θ + n θ + n+ = θ + n cos(α), ω < θ + n <. cos(α) (θ n + ) 4 g sin α sin γ, θ + w Notice that the formery undefined points at {ω, ω } are now we-defined transitions with ω =, because it is kinematicay impossibe to have the whee staticay baancing on a singe eg Fixed Points and Stabiity For a fixed point, we require that θ + points, depending on the parameters: n = θ + = ω. Our system has two possibe fixed n+ n ω =, ω = cot(α) 4 g stand ro sin α sin γ. The imit cyce potted in Figure 5.4 iustrates a state-space trajectory in the vicinity of the roing fixed point. ωstand is a fixed point whenever γ < α. ωro is a fixed point whenever ωro > ω. It is interesting to view these bifurcations in terms of γ. For sma γ, ω stand is the ony fixed point, because energy ost from coisions with the ground is not compensated for by gravity. As we increase γ, we obtain a stabe roing soution, where the coisions with the ground exacty baance the conversion of gravitationa potentia to kinetic energy. As we increase γ further to γ > α, it becomes impossibe for the center of mass of the whee to be inside the support poygon, making standing an unstabe configuration. c Russ Tedrake, 9

7 48 Chapter 5 Waking 6 ω ω 4 /dt [ang. ve. after coision n+] dθ n+ + 4 ω* stand ω* ro Return map 6 Reference ine of sope Fixed points ω (eft) and ω dθ n + /dt [ang. ve. after coision n] FIGURE 5.5 Limit cyce trajectory of the rimess whee (m =, =, g = 9.8, α = π/8, γ =.5). A hatched regions converge to the roing fixed point, ωro ; the white regions converge to zero veocity (ωstand ). 5.5 THE COMPASS GAIT The rimess whee modes ony the dynamics of the stance eg, and simpy assumes that there wi aways be a swing eg in position at the time of coision. To remove this assumption, we take away a but two of the spokes, and pace a pin joint at the hip. To mode the dynamics of swing, we add point masses to each of the egs. For actuation, we first consider the case where there is a torque source at the hip - resuting in swing dynamics equivaent to an Acrobot (athough in a different coordinate frame). a θ γ sw b τ m m θ st h FIGURE 5.6 The compass gait c Russ Tedrake, 9

8 Section 5.6 The Kneed Waker 49 In addition to the modeing assumptions used for the rimess whee, we aso assume that the swing eg retracts in order to cear the ground without disturbing the position of the mass of that eg. This mode, known as the compass gait, is we studied in the iterature using numerica methods [36, 8], but reativey itte is known about it anayticay. The state of this robot can be described by 4 variabes: θ st, θ sw, θ st, and θ sw. The abbreviation st is shorthand for the stance eg and sw for the swing eg. Using q = [θ sw, θ st ] T and u = τ, we can write the dynamics as H(q)q + C(q, q )q + G(q) = Bu, with mb mb cos(θ st θ sw ) H = mb cos(θ st θ sw ) (m h + m) + ma mb sin(θ st θ sw )θ st C = mb sin(θ st θ sw )θ sw mbg sin(θ sw ) G =, (m h + ma + m)g sin(θ st ) B = and = a + b. These equations come straight out of [37]. The foot coision is an instantaneous change of veocity governed by the conservation of anguar momentum around the point of impact: + Q + (α)q = Q (α)q, where mab mab + (m h + ma) cos(α) Q (α) = mab mb(b cos(α)) m( b cos(α) + ma + m h Q + (α) = mb mb cos(α) θ sw θ st and α =. Numerica integration of these equations reveas a stabe imit cyce, potted in Figure 5.7. The cyce is composed of a swing phase (top) and a stance phase (bottom), punctuated by two instantaneous changes in veocity which correspond to the ground coisions. The dependence of this imit cyce on the system parameters has been studied extensivey in [37]. The basin of attraction of the stabe imit cyce is a narrow band of states surrounding the steady state trajectory. Athough the simpicity of this mode makes it anayticay attractive, this ack of stabiity makes it difficut to impement on a physica device. 5.6 THE KNEED WALKER To achieve a more anthropomorphic gait, as we as to acquire better foot cearance and abiity to wak on rough terrain, we want to mode a waker that incudes knee[4]. For this, we mode each eg as two inks with a point mass each. c Russ Tedrake, 9

9 5 Chapter 5 Waking.5.5 dθ Pitch /dt θ Pitch FIGURE 5.7 Limit cyce trajectory of the compass gait. (m = 5kg,m h = kg,a = b =.5m,φ =.3deg. x() = [,,,.4] T ). θ P itch is the pitch ange of the eft eg, which is recovered from θ st and θ sw in the simuation with some simpe book-keeping. m H b m t mt q a L b q 3 q a m s m s FIGURE 5.8 The Kneed Waker At the beginning of each step, the system is modeed as a three-ink penduum, ike the baistic waker[59, 58, 8]. The stance eg is the one in front, and it is the first ink of a penduum, with two point masses. The swing eg has two inks, with the joint between them unconstrained unti knee-strike. Given appropriate mass distributions and initia conditions, the swing eg bends the knee and swings forward. When the swing eg straightens out (the ower and upper ength are aigned), knee-strike occurs. The knee-strike is modeed as a discrete ineastic coision, conserving anguar momentum and changing veocities instantaneousy. c Russ Tedrake, 9

10 Section 5.6 The Kneed Waker 5 After this coision, the knee is ocked and we switch to the compass gait mode with a different mass distribution. In other words, the system becomes a two-ink penduum. Again, the hee-strike is modeed as an ineastic coision. After the coision, the egs switch instantaneousy. After hee-strike then, we switch back to the baistic waker s three-ink penduum dynamics. This describes a fu step cyce of the kneed waker, which is shown in Figure ink dynamics -ink dynamics Knee-strike Hee-strike FIGURE 5.9 Limit cyce trajectory for kneed waker By switching between the dynamics of the continuous three-ink and two-ink penduums with the two discrete coision events, we characterize a fu point-feed kneed waker waking cyce. After finding appropriate parameters for masses and ink engths, a stabe gait is found. As with the compass gait s imit cyce, there is a swing phase (top) and a stance phase (bottom). In addition to the two hee-strikes, there are two more instantaneous veocity changes from the knee-strikes as marked in Figure 5.. This imit cyce is traversed cockwise..5 Right Leg.5 dθ/dt.5 switch to swing phase (hee strike) swing eg whie knee strikes.5.5 stance eg whie knee strikes switch to stance phase (hee strike) θ FIGURE 5. The Kneed Waker c Russ Tedrake, 9

11 5 Chapter 5 Waking 5.7 NUMERICAL ANALYSIS A note on integrating hybrid systems and/or evauating return maps. The dynamics are often very sensitive to the switching pane. Often a good idea to back up integration to attempt to find coisions and transitions very accuratey. MATLAB can hande this nicey with zero-crossings in the ODE sovers Finding Limit Cyces You can find asymptoticay stabe imit cyces by integrating forward the ODE (unti t ), as ong as you can guess an initia condition inside the basin of attraction. This convergence rate wi depend on the convergence rate around the fixed point and coud be inefficient for compex systems, and guessing initia conditions is difficut for systems ike the compass gait. This method won t work for finding unstabe imit cyces. Remember that a fixed point on the Poincare map: x p [n + ] = P(x p [n]), is simpy a zero-crossing of the (vector-vaued) function P(x p [n]) x p [n + ]. Therefore, an efficient way to obtain the fixed points numericay is to use an agorithm which finds the zero-crossings by a Newton method[6, 67]. These methods can be dramaticay more efficient if one can efficienty estimate the gradient of the Poincare map. Gradients of the Poincare map. The Poincare map is defined by integrating the continuous dynamics, t c [n+] + + c c c + t c [n] x(t [n + ]) = x(t [n]) + f(x(t))dt, x(t [n]) = x p [n] then appying the (discrete) impact dynamics + c c x p [n + ] = x(t [n + ]) = F(x(t [n + ])), where t c [k] is the time of the kth coision, and t c indicates just prior to the coision and t + c is just after the coision. In order to estimate the gradients of the Poincare map, dx p[n+] dx p[n], we must take a itte care in handing the effects of initia conditions on the time (and therefore the state) of coision (using the chain rue). Linearizing about a trajectory x (t), x p [n] with impacts at t c [n], we have: [ ] dx p [n + ] F(x) x(t) dt c [n + ] = + f(x) dx p [n] x x p [n] dx p [n] t=t c [n+],x=x (t c [n+]) The switching dynamics are defined by the zeros of scaar coision function, φ(t, x). Athough these dynamics are time-varying in genera (e.g., moving obstaces), for the rimess I hand it to SNOPT as a constraint. c Russ Tedrake, 9

12 Section 5.7 Numerica Anaysis 53 whee the coision function can be as simpe as φ(x) = sgn(θ )(θ γ) α. This coision [n+] function aows us to compute dtc : dx p[n] [ ] dφ(t c [n + ], x(t c [n + ]) φ(t, x) dx(t) x(t) dt c [n + ] φ(t, x) dt c [n + ] = = + + dx p [n] x dx p [n] t dx p [n] t dx p [n] c c φ(t,x) x(t) dt c [n + ] x x = p[n] dx φ(t,x) + φ(t,x) p [n] f(x) x(t) The fina step, computing x p[n] t x t=t c [n+],x=x(t c [n+]) t=t c [n+] ation - see section.3. for the update., can be done with a standard gradient cacu t=t [n+],x=x(t [n+]) EXAMPLE 5.3 Fixed points of the rimess whee Bifurcation diagram (as a function of sope) from computationa search. Show that it agrees with anaytica soution. EXAMPLE 5.4 Fixed points of the compass gait Bifurcation diagram (as a function of sope) - computationa ony Loca Stabiity of Limit Cyce In practice, the oca stabiity anaysis of a imit cyce is done by taking the derivatives around the fixed point of the return map. Again, this is often accompished using numerica derivatives. Perturb the system in one direction at a time, evauate the map and buid the matrix... From Goswami [37]. The eigenvaues of the derivative matrix of the Poincaré map, λ i are caed the characteristic or Foquet mutipiers[83]. P x + δ = P(x + δ ) P(x ) + δ. x x P δ δ. x x A fixed point is stabe if the n non-trivia eigenvaues of this matrix are λ i <. Trivia mutipiers vs. Non-trivia mutipiers. Expect one trivia mutipier of, or (which revea the dynamics of a perturbation aong the imit cyce orbit). A standard numerica recipe for estimating is to perturb the system by a very sma amount at east n times, once in each of the state variabes, and watching the response. Be carefu - your perturbation shoud be big enough to not get into integration errors, but sma enough that it stays in the inear regime. A good way to verify your resuts is to perturb the system in other directions, and other magnitudes, in an attempt to recover the same eigenvaues. In genera, the matrix P x can be reconstructed from any c Russ Tedrake, 9 P x

13 54 Chapter 5 Waking PROBLEMS number of samped trajectories by soving the equation δ m P δ δ = δ δ δ m x x in a east-squares sense, where δ i is the i-th perturbation (not a perturbation raised to a power!). Lyapunov exponent. There is at east one quantifier of imit cyce (or trajectories, in genera) stabiity that does not depend on the return map. Like a contraction mapping perturb origina trajectory in each direction, bound recovery by some exponentia[83]. δ(t) < δ()e At. The eigenvaues of A are the Lyapunov exponents. Note that for a stabe imit cyce, the argest Lyapunov exponent wi be (ike the trivia foquet mutipier), and it is the remaining exponents that we wi use to evauate stabiity. 5.. (CHALLENGE) Cosed-form soution for the rimess whee. We have a cosed-form expression for the return map, but can you find a soution for the entire return map dynamics? Given θ [], directy compute θ [n] for any n. This woud invove soving the quadratic difference equation. 5.. (CHALLENGE) Deadbeat contro of the compass gait. Find a hip torque poicy that produces a (ocay) deadbeat controer. c Russ Tedrake, 9

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