784 Part D Manipulation and Interfaces Observation of(33.10)reveals that SNEw S,and there- The above method works well if the system model fore any negative feedback from the exoskeleton leads (i.e.,G)is well known to the designer.If the model is to an even smaller sensitivity transfer function.With re- not well known,then the system performance will differ spect to (33.10),our goal is to design a controller for greatly from the one predicted by (33.13),and in some a given S and G such that the closed-loop response from cases instability will occur.The above simple solution d to v(the new sensitivity function as given by (33.10))is comes with an expensive price:robustness to parameter greater than the open-loop sensitivity transfer function variations.In order to get the above method working, (i.e.,S)within some bounded frequency range.This one needs to know the dynamics of the system well, design specification is given by inequality (33.11) to understand the dynamics of the exoskeleton quite well,as the controller is heavily model based.One can ISNEWI>IS,Vo∈(0，oo), (33.11) see this problem as a tradeoff:the design approach de- or alternatively scribed above requires no sensor (e.g.,force or EMG)in 1+GC1<1Vo∈(0，o0), (33.12 the interface between the pilot and the exoskeleton;one can push and pull against the exoskeleton in any direc- where @o is the exoskeleton maneuvering bandwidth. tion and at any location without measuring any variables In classical and modern control theory,every ef-on the interface.However,the control method requires fort is made to minimize the sensitivity function of a very good model of the system.At this time,the experi- a system to external forces and torques.But for exo-ments with the exoskeleton have shown that this control skeleton control,one requires a totally opposite goal: scheme -which does not stabilize the exoskeleton- to maximize the sensitivity of the closed-loop system to forces the system to follow wide-bandwidth human ma- forces and torques.In classical servo problems,negative neuvers while carrying heavy loads.We have come to feedback loops with large gains generally lead to small believe,to rephrase Friedrich Nietzsche,that that which sensitivity within a bandwidth,which means that they does not stabilize,will only make us stronger.Refer- reject forces and torques (usually called disturbances). ence [33.37]describes a system identification method However,the above analysis states that the exoskeleton for BLEEX. controller needs a large sensitivity to forces and torques. How does the pilot dynamic behavior affect the exo- To achieve a large sensitivity function,it is suggested skeleton behavior?In our control scheme,there is no that one uses the inverse of the exoskeleton dynamics as need to include the internal components of the pilot limb a positive feedback controller so that the loop gain for model:the detailed dynamics of nerve conduction.mus- the exoskeleton approaches unity (slightly less than 1). cle contraction,and central nervous system processing Assuming positive feedback,(33.10)can be written as are implicitly accounted for in constructing the dynamic model of the pilot limbs.The pilot force on the exoskel- S SNEW=d=1-GC (33.13) eton,d,is a function of both the pilot dynamics,H,and the kinematics of the pilot limb (e.g.,velocity,position If C is chosen to be C=0.9G-1,then the new sen- or a combination thereof).In general,H is determined sitivity transfer function is SNEw =10S(ten times the primarily by the physical properties of the human dy- force amplification).In general we recommend the use namics.Here it is assumed H is a nonlinear operator of positive feedback with a controller chosen as: representing the pilot impedance as a function of the pilot kinematics C1-a-lG-1, (33.14) d=-H(w). (33.15) where a is the amplification number greater than unity The specific form of H is not known other than that it (for the above example,a =10 led to the choice results in the human muscle force on the exoskeleton of C=0.9G-1).Equation (33.14)simply states that Figure 33.14 represents the closed-loop system behavior a positive-feedback controller needs to be chosen as when pilot dynamics is added to the block diagram of Part the inverse dynamics of the system dynamics scaled Fig.33.13.Examining Fig.33.14 reveals that (33.13), down by (1-).Note that (33.14)prescribes the representing the new exoskeleton sensitivity function,is controller in the absence of unmodeled high-frequency not affected by the feedback loop containing H. exoskeleton dynamics.In practice,C also includes Figure 33.14 shows an important characteristic of a unity-gain low-pass filter to attenuate the unmodeled exoskeleton control.One can observe two feedback high-frequency exoskeleton dynamics. loops in the system.The upper feedback loop represents784 Part D Manipulation and Interfaces Observation of (33.10) reveals that SNEW ≤ S, and there￾fore any negative feedback from the exoskeleton leads to an even smaller sensitivity transfer function. With re￾spect to (33.10), our goal is to design a controller for a given S and G such that the closed-loop response from d to v (the new sensitivity function as given by (33.10)) is greater than the open-loop sensitivity transfer function (i. e., S) within some bounded frequency range. This design specification is given by inequality (33.11) |SNEW| > |S| , ∀ω ∈ (0, ω0) , (33.11) or alternatively |1+ GC| < 1 ∀ω ∈ (0, ω0) , (33.12) where ω0 is the exoskeleton maneuvering bandwidth. In classical and modern control theory, every ef￾fort is made to minimize the sensitivity function of a system to external forces and torques. But for exo￾skeleton control, one requires a totally opposite goal: to maximize the sensitivity of the closed-loop system to forces and torques. In classical servo problems, negative feedback loops with large gains generally lead to small sensitivity within a bandwidth, which means that they reject forces and torques (usually called disturbances). However, the above analysis states that the exoskeleton controller needs a large sensitivity to forces and torques. To achieve a large sensitivity function, it is suggested that one uses the inverse of the exoskeleton dynamics as a positive feedback controller so that the loop gain for the exoskeleton approaches unity (slightly less than 1). Assuming positive feedback, (33.10) can be written as SNEW = v d = S 1− GC . (33.13) If C is chosen to be C = 0.9G−1, then the new sen￾sitivity transfer function is SNEW = 10S (ten times the force amplification). In general we recommend the use of positive feedback with a controller chosen as: C(1−α−1)G−1 , (33.14) where α is the amplification number greater than unity (for the above example, α = 10 led to the choice of C = 0.9G−1). Equation (33.14) simply states that a positive-feedback controller needs to be chosen as the inverse dynamics of the system dynamics scaled down by (1−α−1) . Note that (33.14) prescribes the controller in the absence of unmodeled high-frequency exoskeleton dynamics. In practice, C also includes a unity-gain low-pass filter to attenuate the unmodeled high-frequency exoskeleton dynamics. The above method works well if the system model (i. e., G) is well known to the designer. If the model is not well known, then the system performance will differ greatly from the one predicted by (33.13), and in some cases instability will occur. The above simple solution comes with an expensive price: robustness to parameter variations. In order to get the above method working, one needs to know the dynamics of the system well, to understand the dynamics of the exoskeleton quite well, as the controller is heavily model based. One can see this problem as a tradeoff: the design approach de￾scribed above requires no sensor (e.g., force or EMG) in the interface between the pilot and the exoskeleton; one can push and pull against the exoskeleton in any direc￾tion and at any location without measuring any variables on the interface. However, the control method requires a very good model of the system. At this time, the experi￾ments with the exoskeleton have shown that this control scheme – which does not stabilize the exoskeleton – forces the system to follow wide-bandwidth human ma￾neuvers while carrying heavy loads. We have come to believe, to rephrase Friedrich Nietzsche, that that which does not stabilize, will only make us stronger. Refer￾ence [33.37] describes a system identification method for BLEEX. How does the pilot dynamic behavior affect the exo￾skeleton behavior? In our control scheme, there is no need to include the internal components of the pilot limb model; the detailed dynamics of nerve conduction, mus￾cle contraction, and central nervous system processing are implicitly accounted for in constructing the dynamic model of the pilot limbs. The pilot force on the exoskel￾eton, d, is a function of both the pilot dynamics, H, and the kinematics of the pilot limb (e.g., velocity, position or a combination thereof). In general, H is determined primarily by the physical properties of the human dy￾namics. Here it is assumed H is a nonlinear operator representing the pilot impedance as a function of the pilot kinematics d = −H(v) . (33.15) The specific form of H is not known other than that it results in the human muscle force on the exoskeleton. Figure 33.14 represents the closed-loop system behavior when pilot dynamics is added to the block diagram of Fig. 33.13. Examining Fig. 33.14 reveals that (33.13), representing the new exoskeleton sensitivity function, is not affected by the feedback loop containing H. Figure 33.14 shows an important characteristic of exoskeleton control. One can observe two feedback loops in the system. The upper feedback loop represents Part D 33.7