3. Modeling and Basic Control 45
3.1.1. Compliantly actuated mechanical system
zvanishes identically.
This property is of major importance in the stability and passivity analysis of gen-eral Euler-Lagrange systems. It holds for any sub-class of the Euler-Lagrange systems introduced in the remainder of this chapter.
Proof. The proof of the above corollary follows directly from Proposition 2.1 and a lemma (see, e. g., [VdS12]) which states that the matrix ˙¯M −2 ¯C is skew symmetric if and only if the equality ˙¯M = ¯C + ¯CT holds.
3.1.1. Compliantly actuated mechanical system
A particular sub-class of under-actuated Euler-Lagrange systems which is in the main focus of the following investigations is given by the so-called compliantly actuated mechanical systems. Thereby, the potential energyU(z) is further specified.
Definition 3.1 (Compliant actuation). The under-actuated Euler-Lagrange system as introduced by (3.3) is said to constitute a compliantly actuated mechanical system if the potential energy
U(z) =Ug(z) +Ue(z) (3.4)
comprises a gravitational potentialUg(z)(which is allowed to vanish on the entire manifold Mm+n) and a positive semi-definite elastic potential
Ue(z)≥0, ∀z∈Rm+n, (3.5)
where for at least one h ∈ {1, . . . , m} and for at least one j ∈ {1, . . . , n}, the ”elastic coupling”,
∂2Ue
∂θh∂qj = ∂2Ue
∂qj∂θh 6= 0, ∂2Ue
∂θh2 >0, ∂2Ue
∂qj2 >0, ∀z∈Rm+n (3.6) is non-zero (and finite).
3.1. Under-actuated Euler-Lagrange systems
θh−1 uh−1
θh uh
qj
qj+1
Figure 3.1.: Compliant actuation and static controllability of the state qj by the control inputuh via θh.
Note that the above definition implies that the elastic potentialUe is such that at least one of the indirectly actuated statesqj is statically controllable via a directly actuated state θh at position level. Loosely speaking, at least one of the actuator inertias is connected at least to one of the link inertias via a spring (cf. Fig. 3.1). An example of a not fully statically controllable system is given by the elastic pendulum, where a point-mass is suspended on a radially acting spring.
In the following, the compliantly actuated system is specified further such that the number of statically controllable indirectly actuated states p equals or is lower than the number of directly actuated states m at position level. Furthermore for the simplicity of the description, it is assumed that all of the indirectly actuated states are statically controllable, i. e., p = n. Then, properties naturally arising for mechanical systems—
when rigid bodies are connected via springs—are assumed. Therefore, basic definitions to evaluate the boundedness of matrices are introduced in advance.
Definition 3.2. Given a square and symmetric real matrix A, then the minimum and maximum eigenvalue ofA are denoted by λmin(A) and λmax(A)≥λmin(A), respectively.
Definition 3.3. Given any real matrix A, then the minimum and maximum singular value of A are denoted by σmin(A) and σmax(A)≥σmin(A), respectively.
Assumption 3.2. The gradient ∂Ue/∂q is strictly monotonic in q in a sense that there exists constantsc1, c2>0 such that
z∈Rinfm+nλmin
∂2Ue(θ,q)
∂q2
> c1, (3.7)
sup
z∈Rm+n
λmax
∂2Ue(θ,q)
∂q2
< c2, (3.8)
keep bounded.
This property ensures that given any fixed θ, the equations1 −(∂Ue/∂q)T = τ have always a unique solution for q. The assumption which ensures that given any fixed q,
−(∂Ue/∂q)T =τ have always a unique solution for θ can be formulated similarly.
1The negative sign is a convention that the generalized elastic forceτ is an ”applied” force.
Assumption 3.3. The gradient ∂Ue/∂q is strictly monotonic in θ in a sense that there
The above property also implies that all of the indirectly actuated states are statically controllable. This in turn means that the generalized force produced by the springs is controllable which is a requirement for the control approaches presented in Sect. 3.3.
Assumption 3.4. The gradient ∂Ue/∂θ is strictly monotonic in θ in a sense that there exists constantsc5, c6>0 such that
The remaining assumptions guarantee the existence of a unique static equilibrium and are therefore imposed on the potential function including gravity.
Assumption 3.5. There exists constantsc7, c8 >0 such that
z∈Rinfm+nλmin
Loosely speaking, condition (3.13) ensures that the compliantly actuated system does not collapse statically under the influence of gravity, if the actuator positions are hold to constant.
Assumption 3.6. There exists constantsc9, c10>0 such that
z∈Rinfm+nσmin
These assumptions suggest the definition of a static equivalent of the indirectly actuated states at position level as introduced in the following.
3.1. Under-actuated Euler-Lagrange systems
Remark 3.1. As a result of the implicit function theorem, the existence of the functions
¯
q= ¯q(θ) is guaranteed due to Assumption 3.5.
Considering an argumentation adopted from [ASOP12] (and ifm=n) it can be shown that ¯q= ¯q(θ) is a diffeomorphism.
Proposition 3.1. In casem=n, the mappingq¯ : Rn→Rn introduced by Definition 3.4 is a global diffeomorphism.
Proof. In order to show that the mapping ¯q : Rn→Rnis a diffeomorphism, the Jacobian
∂¯q(θ)/∂θ needs to be shown to be always nonsingular, i. e., it has to be shown that sup
keeps bounded2 from above [Zei86, Corollary 4.41, p. 174]. By differentiating condition (3.18) it can be seen that the Jacobian matrix of the mapping ¯q takes the form
Jq¯(θ) := ∂¯q(θ)
The inverse of this Jacobian matrix Jq¯(θ)−1 keeps bounded due to Assumptions 3.5 and 3.6.
Remark 3.2. Proposition 3.1 can be extended to the more general case: m > n. This can be accomplished by introducing m−n holonomic constraints φ(θ) = 0, where the constraint Jacobian matrix∂φ(θ)/∂θ is of full rank (cf. Theorem 2.4).
The remaining assumption on the potential U(θ,q) is made mainly for the case of classical actuator position PD control.
Assumption 3.7. There exists a constant c11∈R such that
z∈Rinfm+nλmin
∂2U(θ,q)
∂θ2
> c11. (3.22)
2The matrix normk · kis assumed to be the induced Euclidean norm.
This assumption is not required (for the stability analysis), if instead of the actua-tor configuration, the homeomorphic static equivalent of the indirectly actuated states (cf. Definition 3.4) is considered as control variable.
Remark 3.3 (Variable stiffness actuation). It is worth mentioning that no assumptions have been made yet on the number of directly and indirectly actuated states at position levelm andn, respectively. In particular, the additional degrees of freedom in casem > n can be exploited to alter the characteristics of the generalized spring force, i. e., the relation between a displacement ∆q w. r. t. an equilibrium point q0 and the resulting generalized force
The possibility to independently alter the ”shape” of the elastic potential in addition to its minimum is known as the principle of variable stiffness actuation (VSA), see, e. g., [ASEG+08].
The concept of variable stiffness actuation is a special case of the more general concept of compliant actuation treated here. The former is circumstantial for this thesis and therefore will not be treated in detail.
Finally, for simplicity and for readability, the derivation of the basic joint-level control approaches treated in this thesis assumes the case, where the number of directly actuated statesm equals the number of statically controllable indirectly actuated states p and the number of indirectly actuated statesn, at position level respectively, i. e.,m=p=n.