We propose a QCQP-representable IMU pre-integration factor that enables certifiable estimation with pre-integrated inertial measurements. To the best of our knowledge, this is the first work to directly incorporate IMU pre-integration into certifiable estimation. Inertial sensing is a common and reliable modality in robotics, and incorporating it broadens the practical scope of certifiable estimation. The main challenges are obtaining the required algebraic structure and a sufficiently tight convex relaxation. Standard IMU pre-integration relies on the exponential map, which does not admit an exact polynomial representation. Moreover, obtaining a QCQP formulation requires auxiliary lifting variables, for which the standard semidefinite programming (SDP) relaxation can be loose. We address these issues by deriving an IMU pre-integration factor based on the Cayley map and an explicit set of redundant constraints that tighten the resulting relaxation. To validate the proposed factor, we apply it to certifiable GNSS-IMU smoothing and evaluate it on synthetic and real-world data. The results show that the proposed formulation yields tight relaxations and solves the resulting estimation problems to verified global optimality.
Figures & tables
ID
Identity
Constrains
A1
RkRk⊤=s2I3
rotation rows
B1
Yij(′)wij=0
lifting–bias
B2
(Yij(′))⊤Yij(′)=∥wij∥2I3−wijwij⊤
lifting–lifting
C1
∥Wij∙∥2=∥uij∙∥2
vector lifting
TABLE I: Redundant-constraint subsets used to tighten the SDP relaxation. Here, Yij(′) denotes either Yij or Yij′ , and ∙∈{v,p} .
Fig. 3: Effect of cumulative redundant-constraint subsets on SDP tightness. Numerical rank and certification rate over 840 SDP solves. Each stage includes all constraint subsets shown to its left. Adding B2 substantially tightens the relaxation, while the full set including C1 produces rank-one solutions across the tested noise range.
Sequence
K
M
n
cert.
med [ η ]
max [ η ]
err. [ ∘ ]
EuRoC MH_01
8
1
334
4/4
4.3
6.7
1.05
16
1
646
4/4
1.2
2.2
0.83
16
2
652
4/4
2.9
10
0.88
24
1
958
4/4
1.5
6.8
0.83
24
3
970
4/4
4.0
6.0
1.15
UrbanNav med.
16
2
652
4/4
9.0
39
0.72
TABLE II: Performance on real-world benchmarks using the full redundant-constraint set. Four windows are evaluated for each configuration. Cert. reports the number of rank-one certified windows. The median and maximum relative gaps are reported in units of 10−6 , and err. is the median window-level mean orientation error.
rot. [degree]
pos. [m]
Method
init
med
max
med
max
cert.
EuRoC MH_01 , 4 windows
Ours (SDP)
random
0.88
7.62
0.018
0.021
4/4
LM [ 10 ]
truth
0.88
7.73
0.019
0.023
—
LM [ 10 ]
random
4.31
179.28
0.019
0.247
—
UrbanNav medium urban , 2 windows
TABLE III: Comparison with standard pre-integration . Medians and maxima are taken across the windows of each dataset; bold marks the best entry of a column. Cert. counts windows returned with a rank-one certificate.
Dynamic Legged Systems (DLS) Lab, Istituto Italiano di Tecnologia, 16163 Genova, Italy · Institut de Rob`otica i Inform`atica Industrial - CSIC, 08028 Barcelona, Spain