2.1.1. Inverse kinematics of parallel manipulator
Let the posture vector of the moving platform beθ=?x,?y,?z. Define
??TPAi?R3as the position vector of the ith spherical joint of the base platform in frame Oxyz and as PBi?R3the position vector of the ith spherical joint of the moving platform in frame O1x1y1z1. Let Th?R3be the position vector of the center O1in frame Oxyz andPBithe position vector of the ith spherical joint of he moving platform in frame Oxyz. Then,
PBi????RO???Pbi+Th （1） where RO???is the rotational matrix from frame O1x1y1z1to frame Oxyz and was uniquely determined using the posture vector? by Huang, Kou, and Fang (1997)
?c?zc?y?R0?????s?zc?y??s?y?c?zs?ys?x?s?zc?xs?zs?ys?x?c?zc?xc?ys?xc?zs?yc?x?s?zs?x??s?zs?yc?x?c?zs?x? （2）
?c?yc?x?where c? and s? are short-hand notations for cos?and sin?respectively. The contractive length of the ith pneumatic muscle xmican then be determined as
xmi????L0?LJi?PAi?PBi(?),i?1,2,3 （3）
where L0is the initial length of pneumatic muscle, Ljiis the ineffective length between the base platform and the moving platform excluding the effective length of the ith pneumatic muscle, and | ? | represents the Euclidean norm of a vector. Eq.(3) solves the inverse kinematics of the moving platform from the posture vector _ to the contractive length vector of three pneumatic musclesXm??Xm1,Xm2,Xm3?.
T2.1.2. Dynamics of parallel manipulator
Let Fi be the value of the force acting on the moving platform by the ith pneumatic muscle along its axial direction, with positive value for the contractive force. Denote a force vector as F=?F1,F2,F3?. Let the angular velocity of the moving platform
Texpressed in the base frame Oxyz be??R3. Then, according to the dynamic equation of a rigid body, the dynamic model of the moving platform can be written as Tao et al. (2005)
~???C?????C?,??t??JT???F (4) I????s????represents the Coriolis terms, where I（?） is the rotational inertia matrix and C?,?Cs?diag?cs1,cs2,cs3?represents the coefficient matrix of viscous frictions of the
T????~~~~Tspherical joints, ?(t)??x,?y,?zrepresents the external disturbance vector, and ??J(?)is the Jacobian transformation matrix between the contractive velocity vector of pneumatic muscles and angular velocity vector of the moving platform.
The dynamic equations of the driving units can be written as
?m?Ff?F?Fm （5） M?xWhere M=diag（?m1,m2,m3?）is the equivalent mass matrix of three pneumatic
Tmuscles and their spherical joints, Ff=Ff1,Ff2,Ff3 is the friction force vector of pneumatic muscles,and Fm??Fm1,Fm2,Fm3?is the static force vector of pneumatic
T??Tmuscles detailed in Section 2.2. Merging Eqs. (4) and (5) while noting
??G?1(?)? in which （G??J(?)?and ?˙X（?）is the transformation matrix from
mthe angular velocity vector to the posture velocity vector of the PMDPM via three RPY angles, the dynamic model of PMDPM can be obtained as follows in terms of posture vector.
???B??????d?,??,t?JT???F （6） IP????Pt?m???GT???I????JT???MJ???G???,B????GT???CG???,d?,??,t? whereIP????Pst??,?????GT???JT???MJ???G?,?????GT???C?,??G?????+ GT???I???G????????????~?G??????GT???JT???F?GT?????t?,J?????J???G???. GT???JT???MJ?,?f??
2.2. Models of actuator
2.2.1. Characteristics of pneumatic muscle
For each driving unit i, the static force of pneumatic muscle is(Chou & Hannaford, 1996; Tondu & Lopez, 2000; Yang, Li, & Liu,2002)
Fmi(xmi,pi)?pia?1?k??mi??b?Fri?xmi???Fi （7）
2???1?12? Fri?xmi???D0L0?0E?1??mi?????sin??cos?0 （8）??sin?x0imi??where piis the pressure inside the pneumatic muscle, a and b are constants related to the structure of pneumatic muscle, k?is a factor accounting for the side effect, ?miis the contractive ratio given by?mi=xmi/L0, Fri(xmi) is the rubber elastic force, D0 is the initial diameter of the pneumatic muscle, ?0is the thickness of shell and bladder,
E is the bulk modulus of elasticity of rubber tube with cross-weave sheath, ?0is the initial braided angle of the pneumatic muscle, ?i(xmi)is the current braided angle of the pneumatic muscle given by ?i(xmi)?arccos??L0?Xmi?cos?0/L0?and ?Fiis the modeling error.
2.2.2. Actuator dynamics
The pressure dynamics that generates the pressure inside the pneumatic muscle will be given. Suppose the relationship between volume and pressure inside the pneumatic muscle can be described by polytropic gas law
?Vi?xmi??pi?（9） ?m???const
ai???iwhere Vi(xmi)is the pneumatic muscle’s inner volume, which is a function of
xmi,mai is the air mass inside the pneumatic muscle, and?i is the polytropic exponent. The ideal gas equation describes the dependency of the air mass.
piVi?xmi??maiRTi （10）where R is the gas constant and Ti is the thermodynamic temperature inside the pneumatic muscle.
Differentiate Eq. (9) while noting Eq. (10), one obtains
?i?p?ai?iRTimVi?xmi????x,x?mi??ipiVimiVi?xmi? （11）
?aiis the air mass flow rate through the valve. where m~(t)in which qmi is the calculable air mass flow rate ??cq?qSuppose maiqmimi~?t? are the coefficient and disturbance considering given in Section 2.2.3, cqand qmimodel errors respectively.Then the following general pressure dynamic equation instead ofEq. (11) will be used (Richer & Hurmuzlu, 2000)
?i?p?biRTiqmiVi?xmi????x,x?mi??aipiVimiVi?xmi? ?dqi?t? （12）
where ?ai??i,?bi?Cq?i,dqi?t??~?t??iRTiqmiVi?xmi?
2.2.3. Valve model
The relationship between the calculable air mass flow rate qmiinto the ith
pneumatic muscle and control inputui (the duty cycle of two fast switching valves in the ith driving unit) can be put into the following concise form (Tao et al., 2005)
qmi?ui??Kqi?pi,sign?ui??ui （13）wherekqi is a nonlinear flow gain function given by
kqi??pisign?ui??=AeiPuiRTui?pdi?f??p??with ?ui?k?1?pdi?2?k?1?k0??0.528???pui?k?1??2k?1????pdi??2k??pdi?k?pdi?k?pdi?????f???0.528??1 ?p??k?1??p??p??pui?ui???ui??ui??????????in which Aeiis the effective orifice area of fast switching valve, pdiis the upstream pressure, pdi is the downstream pressure, Tuiis the upstream temperature and k is the ratio of specific heat.
2.3. Dynamics in state-space form
Let Fm?xm,p??Fm?xm,p???Fbe the calculable and differentiable part of Fm. And define the drive moment of the PMDPM in task-space as
T???Fm?xm,P? （14） ??J?Differentiate Eq. (14) while noting Eqs. (12) and (13), the dynamics of actuators are described as
?,p??g???K?p,sign?u??u?d?t? （15）??f???,? ??q?Where
?,p?JT?,??F?x,p??JT????Fm?xm,p?x?m- f??,??mm??xm????T???J??Fm?xm,p??m,p??aAf?xm,x?m,p? Af?xm,x?pT??pV?X,X???pV??x,x??X,x?????pV11M1m122m2m233m3m3?? =diag?,,????V1?xm`?V2?xm2?V3?xm3????T???g?????J??Fm?xm,p?Ag?xm?diag??b? ?pT??RT??RT3RT21?? Ag?xm??diag?,,???V1?xm1?V2?xm2?V3?xm3?????T???d??t??J??Fm?xm,p?Tdq?t?,dq?dq1,dq2,dq3 ?p??T????Fm?xm,p??Fm1,p1?Fm2,p2?Fm3,p3??? ?diag?,,????p1?p?p2?p3???????F?x,p??F?x,p??F?x,p??T??Fm?xm,p??diag??m1m11,m2m22,m3m33??
???xm?xm1?xm2?xm3????kq?p,sign?u???diagkq1?p1,sign?u1??,kq2?p2,sign?u2??,kq3?p3,sign?u3??????
TTTTTu??u1,u2,u3?,p??p1,p2,p3?,?a???a1,?a2,?a3?,?b???b1,?b2.?b3?
Define a set of state variables as x=x1x2,x3?T,TTT????T?T,?T,??.Then the entire
Tsystem can be expressed in a state-space form as
?1?x2??x???1?2?IP?x1?x3?Bp?x1?x2?dpx（16） ?? ?x?????????fx,x,fx,x?gxku?d3?12p13?1q?????wherefp?x1,x3? is the inverse function of Eq. (14) from ?to
?,t?JT????. Note that both I?x?and g?x?are positive p,anddp?dt?,??1p1?F??definite matrices. For the simplicity of notation, the variables such as J????and
Ip???are expressed by J?andIp when no confusions on the function variables exist.
3. Adaptive robust controller
3.1. Design issues to be addressed
Generally the system is subjected to parametric uncertainties due to the variation of cs,Ip,?a,?betc., and uncertain nonlinearities represented by dpand dithat come from uncompensated model terms due to complex computation, unknown model