In　this　paper,　a　novel　trajectory　tracking　control　method　of　nonholonomic　mobile　robots　based　on　the　non-negative　piecewise　predefined-time　theorem　is　proposed.　The　idea　of　cascade　control　is　used　to　divide　the　posture　error　system　of　the　mobile　robot　into　two　subsystems.　Firstly,　for　the　first-order　subsystem,　an　active　predefined-time　controller　is　designed　to　realize　that　the　angle　error　system　converges　and　stabilizes　to　zero　within　a　given　time,　which　is　preset　in　advance.　Secondly,　a　novel　predefined-time　sliding　mode　controller　is　designed　for　the　second-order　subsystem,　which　adds　a　constant　to　compensate　for　the　influence　of　singularity.　Moreover,　compared　with　the　existing　fixed-time　control　algorithm,　the　control　scheme　proposed　in　this　paper　provides　a　more　accurate　upper　bound　of　the　settling　time　estimation.　For　convenience,　the　complex　expression　of　the　settling　time　estimation　is　transformed　into　an　adjustable　parameter.　Furthermore,　the　stability　of　the　two　developed　controllers　is　analyzed　and　some　conditions　for　selecting　parameters　are　given.　Finally,　the　simulation　results　show　the　feasibility　and　correctness　of　the　proposed　control algorithm.　©　2021　The　Authors.　IET　Control　Theory　&　Applications　published　by　John　Wiley　&　Sons　Ltd　on　behalf　of　The　Institution　of　Engineering　and　Technology