## I Introduction

Many important control problems can be formulated in terms of two concurrent objectives: the need to reach a given *goal* point while staying inside some *safe* part of the statespace. This applies to autonomous cars, drones and other unmanned vehicles, but also to e.g. a manipulator arm grasping an object while staying away from joint limits as well as workspace obstacles.

Lyapunov theory [12] is a well known tool used to address the first of these objectives, goal convergence (including stability). However, it turns out that the same ideas of Lyapunov can also be used to address the second objective, staying within a safe set. This is the key observation behind two approaches that were developed independently, Behavior Control Lyapunov Functions (BCLFs) [9] and Control Barrier Functions (CBFs) [1].

BCLFs [9], were inspired by the ideas of handling concurrent objectives in the subsumption architecture [6] and the behavior based approach [4], but strived to do so in a more formal way, with guarantees on performance. This was done by applying the ideas of Control Lyapunov Functions (CLFs) [12], but having several such functions in parallel, and placing bounds on the their time derivatives that depended on the available margins.

Allowing for several concurrent objectives, BCLFs also include mechanisms for different priority levels, and acceptable safety margins for each of these levels and objectives. However, picking just one additional objective, and one priority level, the key result of BCLFs are very similar to those of CBFs, as will be shown in this paper.

CBFs were introduced in [2], and have since been applied to an ever increasing number of applications [11, 10, 3, 14, 7, 8].

The main contribution of this paper is that we show that there are significant similarities between the theory of CBFs and BCLFs. These similarities have not been shown before. In fact, the control community has been largely unaware of BCLFs, as can be seen from the detailed description of the history of CBFs included in [1]. There it is stated that the first control version of a barrier certificate was presented in 2007, [13], and that the notion of extending at the boundary of the safe set to for the entire set was first done in 2009, [5]. As shown below, both of these features were described in 2006, in [9], applying the constraint for some , which is similar in spirit to the more general that was introduced in [2, 3].

The outline of this paper is as follows. First we provide a background description of the key results of CBFs and BCLFs in Section II. Then we show the similarity of the two results in Section III, followed by conclusions in Section IV.

## Ii Background

In this section we will revise some key results on CBFs as well as BCLFs. For a complete description we refer the reader to [1] and [9].

### Ii-a Control Barrier Functions

In [1], we have the following definitions and results.

###### Definition 1 (From [1])

Let be the superlevel set of a continuously differentiable function , then is a control barrier function (CBF) if there exists an extended class function such that for ,

(1) |

for all .

Let the set of control values that render the set safe be defined as

(2) |

###### Theorem 1 (From [1])

Let be a set defined as the superlevel set of a continuously differentiable function . If is a control barrier function on and for all , then any Lipschits continuous controller renders the set safe. Additionally, the set C is asymptotically stable in .

### Ii-B Behavior Control Lyapunov Functions

The ideas presented in [9] is to define not a single safe set as in [1] but a family (over and ) of such sets . Here indicates different types of objectives, such as obstacle clearance or time at target, whereas indicates different levels of ambition, such as having an obstacle clearance of 10 or 20 inches.

The initial definition of BCLFs is a bit different from Definition 1, since we are aiming for systems that can achieve objectives that are not met as well as keep those that are already met.

###### Definition 2 (III.1 in [9])

Given a system , a bounded set of admissible controls a piecewise function , and scalars . Then is a BCLF for the bound and if

(3) |

In order to keep track of the family of objectives, over and be able to make informed tradeoffs between them, we need to define a priority level, as illustrated in Table I and defined below.

CPL 0 | CPL 1 | CPL 2 | CPL 3 | ||
---|---|---|---|---|---|

Obstacle separation | 10in | 10in | 20in | ||

Time at target | 60s | 60s | |||

(other objective) | 100 |

###### Definition 3

Note that demanding , for all guarantees that the CPL is always defined. Next comes the BCLF set corresponding to for CBFs.

###### Definition 4 (III.4 in [9])

Fix and let be the same as in Definition 2. Given a set of BCLFs, , with corresponding bounds, . Let

(5) |

the set of controls satisfying the bounds of the CPL. Let furthermore

(6) |

the set of objectives to be focused on and

(7) |

the set of controls aiming to increase the CPL.

Note that is the set of controls preserving the CPL, are the indices of the bounds that are not satisfied at the next CPL, and is that set of controls that move the state towards satisfy those bounds.

More formally, we characterize the sets that guarantee the satisfaction of the CPL and the future increase in priority level as follows.

###### Lemma 1

(III.5 in [9]) If a system starts at , and the chosen controls u satisfy

(8) |

then i.e. the CPL will not decrease. If furthermore

then will be satisfied in finite time, i.e. the CPL will increase.

See [9].

Finally, the asymptotical stability part of Theorem 1 has its counterpart below.

###### Remark 1

(III.6 in [9]) The constant governs how fast a satisfied bound can be approached. If the worst possible is chosen, then we have equality in the constraints, , and approaches exponentially, with the time constant .

## Iii Similarity of Results

The key similartiy between [9] and [1] is the set of controls making the set invariant, in Equation (2) and in Equation (5).

###### Lemma 2

The key sets of control choices that guarantee satisfaction of the objective, and are identical, if the class K function is chosen to be a generic linear function for some , and the family of sets in (4) are reduced to a single one, that is , with .

Having , , the user defined bound , and the class k function for some user defined constant we have that

(9) |

With this observation made, Lemma III.5 of [9] states that implies that the CPL will not decrease, which means that for a given , which translates to .

Therefore, Lemma III.5 is very similar to Theorem 2, stating that implies . Finally, the asymptotic stability of the safe set is also noted in Remark III.6.

## Iv Conclusion

In this paper we have shown the significant overlap between the CBF and the BCLF. This is important not only for understanding how ideas have developed, but also for future work on extending CBFs to domains where more than two objectives are concurrently considered.

## References

- [1] (2019) Control barrier functions: theory and applications. arXiv preprint arXiv:1903.11199. Cited by: §I, §I, §II-A, §II-B, §II, §III, Definition 1, Theorem 1.
- [2] (2014) Control barrier function based quadratic programs with application to adaptive cruise control. In 53rd IEEE Conference on Decision and Control, pp. 6271–6278. Cited by: §I, §I.
- [3] (2016) Control barrier function based quadratic programs for safety critical systems. IEEE Transactions on Automatic Control 62 (8), pp. 3861–3876. Cited by: §I, §I.
- [4] (1998) Behavior-based robotics. MIT press. Cited by: §I.
- [5] (2009) Viability theory. Springer Science & Business Media. Cited by: §I.
- [6] (1986) A robust layered control system for a mobile robot. IEEE journal on robotics and automation 2 (1), pp. 14–23. Cited by: §I.
- [7] (2020) Reinforcement learning for safety-critical control under model uncertainty, using control lyapunov functions and control barrier functions. arXiv preprint arXiv:2004.07584. Cited by: §I.
- [8] (2020) Control barrier functions for nonholonomic systems under risk signal temporal logic specifications. In 2020 59th IEEE Conference on Decision and Control (CDC), pp. 1422–1428. Cited by: §I.
- [9] (2006) Autonomous ucav strike missions using behavior control lyapunov functions. In AIAA Guidance, Navigation, and Control Conference and Exhibit, pp. 6197. Cited by: §I, §I, §I, §II-B, §II-B, §II, §III, §III, Definition 2, Definition 3, Definition 4, Lemma 1, Remark 1.
- [10] (2017) The robotarium: a remotely accessible swarm robotics research testbed. In 2017 IEEE International Conference on Robotics and Automation (ICRA), pp. 1699–1706. Cited by: §I.
- [11] (2014) Uniting control lyapunov and control barrier functions. In 53rd IEEE Conference on Decision and Control, pp. 2293–2298. Cited by: §I.
- [12] (2013) Nonlinear systems: analysis, stability, and control. Vol. 10, Springer Science & Business Media. Cited by: §I, §I.
- [13] (2007) Constructive safety using control barrier functions. IFAC Proceedings Volumes 40 (12), pp. 462–467. Cited by: §I.
- [14] (2019) Decentralized merging control in traffic networks: a control barrier function approach. In Proceedings of the 10th ACM/IEEE International Conference on Cyber-Physical Systems, pp. 270–279. Cited by: §I.

Comments

There are no comments yet.