Exponential Control Barrier Functions or Enorcing High Relative-Degree Saety-Critical Constraints Quan Nguyen and Koushil Sreenath Abstract We introduce Exponential Control Barrier Functions as means to enorce strict state-dependent high relative degree saety constraints or nonlinear systems. We also develop a systematic design method that enables creating the Exponential CBFs or nonlinear systems making use o tools rom linear control theory. The proposed control design is numerically validated on a relative degree 6 linear system (the serial cartspring system) and on a relative degree nonlinear system (the two-link pendulum with elastic actuators.) I. INTRODUCTION Saety-critical constraints are dynamical constraints that require the orward-invariance o a sae set, deined as super-level sets o scalar constraint unctions. Control design to enorce these strict constraints or nonlinear systems is challenging. Enorcing constraints with high relative degree is harder. A ew control techniques to handle input and state-based constraints involve Model Predictive Control (MPC) [9], LMI optimizations [6], reerence governors [], [8], optimal control [], [5], reachability analysis [], [5], Barrier unctions [7], [5], [6], [9], and other approaches [8], [6], []. However these methods typically involve one or more o the ollowing shortcomings: not applicable to nonlinear systems, not applicable or real-time control, incapable o handling constraints with higher relative degree. More recently, Control Barrier Functions (CBFs) [] were introduced to convert saety constraints into a statedependent linear inequality constraints on the inputs or application to adaptive cruise control. This method has been extended to general Riemannian maniolds in []. This approach combines control Lyapunov unctions (CLFs) or stability [] and CBFs or saety and solves a statedependent quadratic program (QP) pointwise in time or the control input []. Combined control Lyapunov-Barrier unctions have also been created by uniting CLFs and CBFs []. To address saety constraints with higher relative degree, the method o control Barrier unctions was extended to position-based constraints with relative degree in [], []. Furthermore, a backstepping based method to design CBFs with higher relative degree was also introduced in [7]. However, achieving a backstepping based CBF design or Q. Nguyen is with the Department o Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 5, email: quannguyen@cmu.edu. K. Sreenath is with the Depts. o Mechanical Engineering, The Robotics Institute, and Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 5, email: koushils@cmu.edu. This work is supported in part by NSF grants IIS-5655, CMMI- 58869, IIS67 and in part by the Google Faculty Research Award. higher relative degree systems (greater than ) is challenging and has not been practically demonstrated. Here we present an alternate design method to address high relative degree constraints. Our method oers a simpler design process, compared to [7], or creating the control barrier unctions or arbitrary high relative degree constraints. Moreover, as we will see, our method is a generalization o the preliminary approach in []. In this paper, we present a novel method called "Exponential Control Barrier Functions" (ECBFs) that can handle state-dependent constraints eectively or nonlinear systems with any relative degree. The design is based on the properties o linear control theory and thereore conventional methods such as pole placement control can be used to design ECBF constraints. The main contributions o the paper with respect to prior work are as ollows: Introduction o Exponential Control Barrier Functions or enorcing high relative degree saety constraints. Formal construction o the Exponential CBFs based on techniques rom linear control theory. Numerical validation o the Exponential CBFs to enorce saety constraints on a nonlinear system with relative degree (the two-link pendulum with elastic actuators), and on a linear system with relative degree 6 (the serial cart-spring system). The rest o the paper is organized as ollows. Section II revists Control Barrier Function and Control Lyapunov Function based Quadratic Programs (CBF-CLF-QPs). Section III introduces the proposed method o Exponential Control Barrier Functions. Section IV presents numerical validation on dierent applications. Finally, Section V provides concluding remarks. II. CONTROL LYAPUNOV FUNCTION AND CONTROL BARRIER FUNCTION BASED QUADRATIC PROGRAMS A. Model REVISITED Consider the nonlinear control aine model ẋ = (x) + g(x)u, () y = h(x), () where x R n is the system state, u R m is the control input, y R m is the control output. Let r be the relative degree o control output y, we have: y (r) = L r h(x) + L g L r h(x)u. ()
The system is Input-Output linearizable i the decoupling matrix L g L r h(x) is invertible, and we can apply an Input- Output linearizing controller: u(x, µ) = u + (L g L r h(x)) µ, () u (x) = (L g L r h(x)) L r h(x). (5) Deining the transverse variable h(x) L h(x) η = L h(x)..., (6) h(x) L r the linearized system becomes: η = + ḡµ, with = F η, ḡ = G, (7) O I... O O O I.. O F =... O.... I O.... O O and G =.... (8) I B. Control Lyapunov Function based Quadratic Programs A control approach based on control Lyapunov unctions, introduced in [], provides guarantees o exponential stability or the traverse variables η. In particular, a unction V (η) is an exponentially stabilizing control Lyapunov unction (ES- CLF) or the system () i there exist positive constants c, c, λ > such that c η V (η) c η, (9) V (η, µ) + λv (η). () Choosing a quadratic CLF candidate, V (η) = η T P η, its time derivative can be computed as V (η, µ) = L V (η) + LḡV (η)µ, () L V (η) = η T (F T P + P F )η, LḡV (η) = η T P G. () We can ormulate the CLF condition in () into a quadratic program (QP) to solve or the control input pointwise in time. This also enables the incorporation o additional constraints. The CLFbased quadratic program (CLF-QP) [] is as ollows: CLF-QP: µ =argmin µ T µ + pd () µ,d s.t. ψ (η) + ψ (η) µ d (CLF) A c (x)µ b c (x) C. Control Barrier Function ψ (η) := L V (η) + λv (η), ψ (η) := LḡV (η). () Consider an aine control system () with the goal to design a controller to keep the state x in the set C = {x R n h(x) }, (5) where h : R n R is a continuously dierentiable unction. Then a unction B : C R is a Control Barrier Function (CBF) [] i there exists class K unctions α and α such that, or all x Int(C) = {x R n : h(x) > }, α (h(x)) B(x) α (h(x)), (6) Ḃ(x, u) = L B(x) + L g B(x)u γ B(x). (7) From [], the important properties o the CBF condition in (7) is that i there exists a Control Barrier Function, B : C R, then C is orward invariant, or in other words, i x() = x C, i.e., h(x ), then x = x(t) C, t, i.e., h(x(t)), t. We can then incorporate the CBF condition into the CLF-QP controller as ollow []: CBF-CLF-QP: where µ =argmin µ T µ + pd (8) µ,d s.t. ψ,ε (η) + ψ,ε (η) µ d (CLF) ψ b (x) + ψ b (x) µ A c (x)µ b c (x) (CBF) ψ(x) b := L B(x) + L g Bu (x) γ B(x), (9) ψ(x) b := L g B(x)(L g L r h(x)), with u (x) as deined in (5). III. EXPONENTIAL CONTROL BARRIER FUNCTION Having revisited control Lyapunov and control Barrier unction based quadratic programs, in this section, we introduce Exponential Control Barrier Functions (ECBFs) and present a novel method to systematically design a exponential CBF or high relative degree constraints. The term Exponential CBF is used since the resulting CBF constraint is an exponential unction o the initial condition. Furthermore, the design and enorcement o ECBFs is based on linear control theory, and as a result, we can easily take advantage o conventional linear control design techniques such as pole placement. Deinition : (Exponential Control Barrier Function): Consider the dynamical system () and the set C = {x R n B(x) }, where B : R n R has relative degree
r b. B(x) is an exponential control Barrier unction (ECBF) i there exists K b R r b s.t., in u U [Lr b B(x) + L gl r b B(x)u + K b η b (x)], x C, () and B(x(t)) C b e Abt η b (x ), when B(x ), the matrix A b is dependent on the choice o K b, and η b (x) = B(x) Ḃ(x) B(x). B (r b ) (x) = B(x) L B(x) L B(x) L r b, (). B(x) C b = [ ]. () Remark : Note that K b, and consequently A b, need to satisy certain properties. We will develop these properties later in this section. For the simple case o relative degree ECBFs, the above deinition can be reormulated as the ollowing: Deinition : (Relative degree Exponential Control Barrier Function): Consider the dynamical system () and the set C = {x R n B(x) }, where B : R n R has relative degree and is continuously dierentiable. B(x) is an exponential control Barrier unction (ECBF) with relative degree i there exists k b R s.t., in [L B(x) + L g B(x)u + k b B(x)], x C, () u U and B(x(t)) B(x )e k bt, when B(x ). Remark : Note that with relative degree, K b and A b in Deinition become scalar k b, a b and the condition or B(x) to be an Exponential CBF is a b = k b >. Remark : (Relation between Exponential CBF and Zeroing CBF): The relative degree ECBF is a Zeroing CBF (ZCBF), as deined in [] since k b B(x) is a class K unction o B(x), and thus retains all the robustness properties o the ZCBF, as detailed in [, Sec..]. We will next introduce the notion o Virtual Input-Output Linearization ollowed by the design o an Exponential CBF. A. Virtual Input-Output Linearization As mentioned in Section II, CLFs are an eective tool to handle stability or both linear and nonlinear systems. Furthermore, there is a systematic way to design CLFs or regulating outputs with arbitrary relative degree r. I we can derive the CBF to the same orm as a CLF, by using another Input-Ouput linearization or the CBF (7), we can then develop a general CBF or constraints with arbitrary relative degree r b. However, input-output linearizing Ḃ(x) is not directly easible due to: (a) the decoupling matrix (L g B(x) when r b = ) being a vector and obviously not invertible, and (b) the control input u in () being already used to Input-Output linearize the output dynamics resulting in (7). In order to solve this problem, we introduce the notion o Virtual Input-Ouput Linearization (VIOL) where an invertible decoupling matrix is not required and the control input µ is chosen to satisy both the CLF condition () as well as input-output linearize the Barrier dynamics as ollows. For a CBF with relative degree, let s deine a virtual control input µ b as ollows: Ḃ(x, µ) = L B(x) + L g B(x)u(x, µ) =: µ b (x, µ). () Note that µ b is a scalar. The CBF condition (7) then simply becomes: µ b (x, µ) γ B(x). (5) We can then let a QP compute µ, µ b so as to simultaneously satisy the CLF condition () as well as both () and (5): CBF-CLF-QP: µ =argmin µ T µ + p d (6) µ,µ b,d s.t. V (η, µ) + λv (η) d (CLF) µ b γ B(x) (CBF) A c (x)µ b c (x) Ḃ(x, µ) = µ b. (VIOL) Remark : Note that the solutions o the two controllers (6) and (8) are identical. However, the VIOL in the above CBF-CLF-QP opens up an eective and systematic way o designing the exponential CBFs. B. Designing Exponential Control Barrier Functions Consider the closed set C = {x R n B(x) }, with our control goal being to ind an input u that guarantees orward invariance o C, i.e., i x := x() C then ind u s.t., B(x(t)), t. ) Simple Case: We irst consider the problem o B(x) having relative degree. Using VIOL, we have, Ḃ(x, µ) = µ b. (7) I we want to drive B(x) to zero, we can simply apply µ b = k b B(x), k b > (8) = B(x(t)) = B(x )e k bt as t. Making use o this property, we can guarantee the inequality constraint B(x) by imposing: µ b k b B(x), k b > (9) = Ḃ(x, µ) k bb(x) () = B(x(t)) B(x )e k bt when B(x ). () Now, we can develop this approach or the general case with B(x) having relative degree r b.
) General Case: Let r b be the relative degree o B(x). We have, B (r b) (x, µ) = L r b B + L gl r b B u(x, µ) = L r b B + L gl r b B ( ) u (x) + (L g L r ) h(x)µ, () where we substitued or u(x, µ) rom (). Further, using VIOL we have B (r b) (x, µ) = µ b, () such that the Input-Output linearized system becomes { η b (x) = F b η b (x) + G b µ b B(x) = C b η b (x), () where η b (x) is as deined in (), F b R r b r b, G b R r b are as deined below..... F b =......., G. b =.., (5).... and C b is as deined in (). From this controllable canonical orm, i we want to drive B(x) to zero, we can easily ind a pole placement controller µ b = K b η b with all negative real poles p b = [ p p... p rb ], with pi >, i =,, r b, that obtains the closed loop matrix A b = F b G b K b with all negative real eigenvalues. Motivated by (8), we can then apply, µ b K b η b, (6) = η b A b η b. (7) Assuming K b = [ kb kb k r ] b b and rom the deinition o η b in (), the last row o the above vector inequality (7) results in B (rb) (x) kb B(x) + kb Ḃ(x) + + kr b b B (rb ) (x). (8) This inequality constraint can also be written in terms o the pole locations p i. To do this, we irst deine a amily o outputs y i : R n R or i =,, r b, as ollows, y i (x) = ( d dt + p ) ( d dt + p ) ( d dt + p r b ) B(x), (9) with y (x) := B(x). Associated with these outputs, we deine a amily o closed sets or i =,, r b, as C i = {x R n y i (x) }. () Remark 5: Note that rom the deinition in (9), the output unctions can also be written recursively as, y i (x) = ẏ i (x) + p i y i (x). () Remark 6: Note that because o choice o K b resulting in poles p b, (8) is identical to y rb (x). Theorem : Suppose the closed-set C rb is orward invariant or the system (), then the closed-set C is orward invariant whenever initially x C i or each i =,, r b, and p i > or each i =,, r b. Beore we prove Theorem, we note the ollowing corollaries: Corollary : Suppose y i (x ) and p i > or i =,, r b, then B(x(t)), t > when B(x ). Proo: This ollows directly rom Theorem and the deinition o the amily o outputs y i in (9) and closed sets C i in (). Corollary : Suppose p i max( ẏi (x) y i (x ), δ), δ >, or i =,, r b, then B(x(t)), t > when B(x ). Proo: This ollows rom Corollary and the act rom () that y i (x ) p i ẏi (x ) y i (x ). Proposition : Suppose the closed-set C i is orward invariant or the system (), then the closed-set C i is orward invariant whenever initially x C i, x C i and p i >. Proo: The proo essentially ollows rom [, Prop. ]. Since x C i, then by orward invariance o C i, we have x(t) C i, or t [, ). From the deinition o C i and (), this is equivalent to ẏ i (x(t)) + p i y i (x(t)), t [, ). Since the trajectory starts in C i, consider the extreme case when the system trajectory reaches the boundary o C i at time T. Then, y i (x(t )) =. However, according to the previous inequality, it ollows that ẏ i (x(t )), which means that the trajectory would never escape C i. We are now ready to prove Theorem. Proo: (o Theorem :) The result ollows rom recursive application o Proposition. In particular, suppose C rb is orward invariant and x C rb, x C rb, p rb >, then rom Proposition, C rb is orward invariant. Further i x C rb and p rb >, then rom Proposition, C rb is orward invariant. We can continue applying Proposition so on to show C is orward invariant. Theorem : (Main Result): Suppose K b is chosen s.t. A b as deined in () is Hurwitz and total negative, and moreover λ i (A b ) ẏi (x) y. Then µ i (x ) b K b η b (x) with η b (x) as in (), guarantees B(x) is a Exponential CBF. Proo: The choice o µ b establishes the invariance o C rb, and K b being Hurwitz and total negative ensures that the eigenvalues o A b are real and negative. The rest ollows rom Corollary and Theorem. In summary, i B(x ) and the designed poles are chosen suiciently small so that the condition in Corollary holds, we can guarantee the state-dependent constraint B(x) by applying the exponential CBF condition (6). Remark 7: (Relation between Exponential CBF and Modiied CBF with position-based constraints []): For saety
Fig. : The Serial Spring Mass System (Relative degree 6). The control goal is to drive the system state rom the initial condition x to a desired position or the rd cart, x d, while strictly enorcing the saety constraint x x max. This system has three degrees-o-reedom and two degreeso-underactuation. constraints with relative degree, the modiication o CBF presented in [] extends the CBF condition or positionbased (relative degree ) constraints o the orm g(x) by enorcing the standard CBF condition [] on h(x) = ( d dt + γ b ) g(x), γ b >. Since, h(x) ġ(x) γ b g(x), enorcing the modiied CBF h(x) results in g(x). g(x) can then be seen as a relative degree Exponential CBF. We now present a QP that incorporates an Exponential CBF into a CLF-QP: ECBF-CLF-QP: argmin µ,δ µ T µ + pδ () s.t. V (η, µ) + λv (η) δ (CLF) A µ C (x)µ bµ C (x) µ b K b η b (Exponential CBF) B (r b) (x, µ) = µ b (VIOL) In the next Section, we will validate our proposed method through two systems: serial spring mass system (linear system with relative degree 6) and a two-link pendulum with elastic actuators (nonlinear system with relative ). IV. SIMULATION RESULT A. Serial Spring Mass System (Relative degree 6) We validate our proposed method on a simple system comprising o serial masses connected through springs as shown in Fig.. The equations o motion or the system is as ollows: m ẍ = u + k(x x ) () m ẍ = k(x x ) + k(x x ) () m ẍ = k(x x ) (5) Deining the system state x = [x x x ẋ ẋ ẋ ] T, we have the linear system: ẋ = Ax + Bu, Fig. : The -link Pendulum with Elastic Actuators (Relative degree ). The control goal is to drive the link angles rom an initial coniguration θ (), θ () to a desired coniguration θ d, θ d while strictly enorcing the saety constraint on the vertical position o the end-eector, p y p min. This system is nonlinear with our degrees-o-reedom and two degreeso-underactuation. A = k m, B = k k m m k m k m k m k m m (6) Our control goal is to drive the system state rom the initial condition x to the desired position x d while considering the saety constraint x x max. As illustrated in Fig., the CLF-QP controller violates the constraint with max(x ) =.7(m) > x max =.5(m), the ECBF-CLF-QP controller with the desired poles p b =. [ 5 ] can handle dierent saety constraints x x max. B. -Link Pendulum with Elastic Actuators (Relative degree ) We consider a two-link pendulum with elastic actuators, as shown in Fig.. The two-link pendulum is a nonlinear system. With elastic actuators, the commanded torques τ, τ o the two motors will generate torques at two joints indirectly through the ollowing motor dynamics: J m θm = k(θ θ m ) + τ, J m θm = k(θ θ m ) + τ, (7) where θ, θ are joint angles o the robot, θ m, θ m are angles o two motors, J m, k are inertia and stiness o motors. Then, the torque at two joints o the robot would be: u = k(θ θ m ) ξ θ, u = k(θ θ m ) ξ θ, (8) where ξ is the damping coeicient at the joints. We apply the Exponential CBF-CLF-QP controller on the two-link pendulum nonlinear system with the above motor dynamics to enorce a constraint on the position o the end eector p (p x, p y ) (see Fig.). While the nominal CLF-QP
[N] [N].5 x x x x x x x x x.5 x x d x max (a) CLF-QP Controller.5.5 x x d x max CBF or x (b) x max x d = 5(cm).5.5 x x d x max CBF or x 6 - (c) x max x d = (cm) Fig. : Serial Spring Mass System: Cart positions and input orce or the Exponential CBF-CLF-QP controller with desired cart position x d = (m) and with saety constraint x x max. The pole locations p b or the CBF encode the perormance speciications o enorcing a saety constraint. As seen above, varying the saety constraint while keeping the poles ixed keeps the peak orces and speed o system response the same. Simulation video: http://youtu.be/okojfutaidk. 6 - u u controller violates the constraint, the ECBF-CLF-QP with poles p b = [ 5 5.5 6 ], can handle dierent saety constraints. C. Discussion Exponential CBFs have the advantage that they can be easily designed or high relative degree saety constraints using tools rom linear control theory. The pole locations or the designed CBF and the poles used to design the CLF can be chosen to more smoothly tradeo stability o tracking and enorcement o saety. Despite these advantages, Exponential Control Barrier Function have some limitations. The choice o pole location depends on initial conditions as stated in Corollary, requiring careul choice o these poles. Although, i the initial conditions are bounded, the poles can be chosen based on these bounds. Furthermore, the presented Exponential CBF-based control design is dependent on the system model and could be sensitive to model uncertainty. Preliminary results to address saety constraints with model uncertainty are presented in []. V. CONCLUSION We have introduced Exponential Control Barrier Functions (ECBFs) as means to enorce high relative degree saety constraints or nonlinear systems. We have presented a systematic design method that enables creating the Exponential CBFs based on pole placement. The designed exponential CBFs along with control Lyapunov unctions (CLFs) were ormulated as a uniied quadratic program. The proposed control design has been numerically validated on a relative degree 6 linear system (the serial cart-spring system) and on a relative degree nonlinear system (the two-link pendulum with elastic actuators.) REFERENCES [] A. D. Ames, J. Grizzle, and P. Tabuada, Control barrier unction based quadratic programs with application to adaptive cruise control, in IEEE Conerence on Decision and Control, Los Angeles, CA, December, pp. 67 678. [] A. D. Ames, K. Galloway, K. Sreenath, and J. W. Grizzle, Rapidly exponentially stabilizing control lyapunov unctions and hybrid zero dynamics, IEEE Transactions on Automatic Control, vol. 59, no., pp. 876 89, Apr.. [] A. Bemporad, Reerence governor or constrained nonlinear systems, IEEE Transactions on Automatic Control, vol., no., pp. 5 9, 998. [] K. Galloway, K. Sreenath, A. D. Ames, and J. W. Grizzle, Torque saturation in bipedal robotic walking through control lyapunov unction based quadratic programs, IEEE Access, vol. PP, no. 99, p., April 5.
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